| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To factor the quadratic expression \(-12 x^2 + 411 x - \frac{4815}{2}\), we will follow these steps: 1. **Identify the coefficients**: The given quadratic is in the form \(ax^2 + bx + c\), where: - \(a = -12\) - \(b = 411\) - \(c = -\frac{4815}{2}\) 2. **Check if the quadratic can be factored easily**: We will first check if the quadratic can be factored by grouping or by finding two numbers that multiply to \(ac\) and add to \(b\). 3. **Simplify the quadratic**: To simplify the factoring process, we can factor out a common factor from each term. Notice that each term has a common factor of 3: \[ -12 x^2 + 411 x - \frac{4815}{2} = 3 \left( -4 x^2 + 137 x - \frac{1605}{2} \right) \] 4. **Factor the simplified quadratic**: Now, we need to factor the quadratic \(-4 x^2 + 137 x - \frac{1605}{2}\). 5. **Find the roots using the quadratic formula**: The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our quadratic: - \(a = -4\) - \(b = 137\) - \(c = -\frac{1605}{2}\) Calculate the discriminant: \[ b^2 - 4ac = 137^2 - 4(-4)\left(-\frac{1605}{2}\right) \] \[ = 18769 - 4 \cdot 4 \cdot \frac{1605}{2} \] \[ = 18769 - 16 \cdot \frac{1605}{2} \] \[ = 18769 - 16 \cdot 802.5 \] \[ = 18769 - 12840 \] \[ = 5929 \] Since the discriminant is a perfect square, we can proceed with factoring: \[ x = \frac{-137 \pm \sqrt{5929}}{2(-4)} \] \[ x = \frac{-137 \pm 77}{-8} \] This gives us two roots: \[ x_1 = \frac{-137 + 77}{-8} = \frac{-60}{-8} = \frac{15}{2} \] \[ x_2 = \frac{-137 - 77}{-8} = \frac{-214}{-8} = \frac{107}{4} \] 6. **Form the factored expression**: The quadratic can be factored as: \[ -4 \left( x - \frac{15}{2} \right) \left( x - \frac{107}{4} \right) \] 7. **Include the factor of 3**: Recall that we factored out a 3 initially, so the final factored form is: \[ 3 \left( -4 \left( x - \frac{15}{2} \right) \left( x - \frac{107}{4} \right) \right) \] \[ = -12 \left( x - \frac{15}{2} \right) \left( x - \frac{107}{4} \right) \] Thus, the factored form of the quadratic is: \[ \boxed{-12 \left( x - \frac{15}{2} \right) \left( x - \frac{107}{4} \right)} \] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| deepseek-r1 | To factor the quadratic expression \(-12x^2 + 411x - \frac{4815}{2}\), we start by eliminating the fraction and simplifying: 1. Multiply the entire quadratic by 2: \[ 2 \left( -12x^2 + 411x - \frac{4815}{2} \right) = -24x^2 + 822x - 4815 \] 2. Factor out \(-1\) from the quadratic: \[ -1 \left( 24x^2 - 822x + 4815 \right) \] 3. Factor out the greatest common divisor (GCD) of the coefficients, which is 3: \[ -1 \cdot 3 \left( 8x^2 - 274x + 1605 \right) \] 4. Factor the quadratic \(8x^2 - 274x + 1605\). Using the quadratic formula, we find the roots: \[ x = \frac{274 \pm 154}{16} \] The roots are \(x = \frac{107}{4}\) and \(x = \frac{15}{2}\). 5. Express the quadratic in factored form: \[ 8x^2 - 274x + 1605 = (4x - 107)(2x - 15) \] 6. Combine the factors and constants, remembering to divide by 2 (since we initially multiplied by 2): \[ -1 \cdot 3 \cdot \frac{1}{2} \left( (4x - 107)(2x - 15) \right) \] 7. Simplify the final expression: \[ -\frac{3}{2} (4x - 107)(2x - 15) \] Thus, the factored form of the quadratic \(-12x^2 + 411x - \frac{4815}{2}\) is: \[ \boxed{-\dfrac{3}{2}(4x - 107)(2x - 15)} \] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| gpto1 | To factor the quadratic expression \( -12x^2 + 411x - \dfrac{4815}{2} \), we start by eliminating the fraction: Multiply both sides by 2: \[ -24x^2 + 822x - 4815 = 0 \] Factor out -1 to make factorization simpler: \[ -1 \left( 24x^2 - 822x + 4815 \right) \] Next, we notice that all the coefficients are divisible by 3: \[ -1 \cdot 3 \left( 8x^2 - 274x + 1605 \right) \] Now, we factor the quadratic \( 8x^2 - 274x + 1605 \). Set up two binomials: \[ (8x - m)(x - n) \] We need \( m \) and \( n \) such that: \[ \begin{cases} mn = 1605 \\ 8n + m = 274 \end{cases} \] Substituting \( m = 1605/n \) into the second equation: \[ 8n + \frac{1605}{n} = 274 \] Multiply both sides by \( n \): \[ 8n^2 + 1605 = 274n \] Bring all terms to one side: \[ 8n^2 - 274n + 1605 = 0 \] Solve for \( n \) using the quadratic formula: \[ n = \frac{274 \pm \sqrt{274^2 - 4 \cdot 8 \cdot 1605}}{2 \cdot 8} \] \[ n = \frac{274 \pm 154}{16} \] We get \( n = 26.75 \) or \( n = 7.5 \) Using \( n = 7.5 \), we find \( m = 214 \) (since \( m = 1605/n \)). So the factors are: \[ (8x - 214)(x - 7.5) \] But since we prefer integer coefficients, we rewrite \( x - 7.5 = x - \dfrac{15}{2} \) and adjust the factors: \[ \begin{align*} (8x - 214)(x - 7.5) &= 2 \left( 4x - 107 \right) \left( x - \dfrac{15}{2} \right) \\ &= \left( 4x - 107 \right) \left( 2x - 15 \right) \end{align*} \] Finally, the original quadratic factors as: \[ -3 \left( 4x - 107 \right) \left( 2x - 15 \right) \] Therefore, the factorization of the original expression is: \(\boxed{-3\left(2x-15\right)\left(4x-107\right)}\) |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| gpt-4o-2024-08-06 | To factor the quadratic \(-12x^2 + 411x - \frac{4815}{2}\), we first multiply the entire expression by 2 to eliminate the fraction: \[ 2(-12x^2 + 411x - \frac{4815}{2}) = -24x^2 + 822x - 4815 \] Now, we need to factor \(-24x^2 + 822x - 4815\). We will use the method of finding two numbers that multiply to \(a \cdot c\) (where \(a = -24\) and \(c = -4815\)) and add to \(b = 822\). First, calculate \(a \cdot c\): \[ a \cdot c = -24 \times -4815 = 115560 \] We need two numbers that multiply to 115560 and add to 822. After testing several pairs, we find that 330 and 351 satisfy these conditions: \[ 330 \times 351 = 115560 \quad \text{and} \quad 330 + 351 = 681 \] However, we made a mistake in the addition. Let's correct it: We need to find two numbers that multiply to 115560 and add to 822. Let's try again: \[ 330 + 351 = 681 \quad \text{(incorrect)} \] Let's try another approach. We need to find the correct pair. After further testing, we find: \[ 330 + 492 = 822 \] Now, we rewrite the middle term using these numbers: \[ -24x^2 + 330x + 492x - 4815 \] Group the terms: \[ (-24x^2 + 330x) + (492x - 4815) \] Factor by grouping: \[ 6x(-4x + 55) + 3(164x - 1605) \] Factor out the common factor: \[ 3(2x(-4x + 55) + 164x - 1605) \] Now, factor the quadratic expression: \[ 3(-4x + 55)(2x - 3) \] Thus, the factorization of the original quadratic is: \[ \boxed{3(-4x + 55)(2x - 3)} \] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| DeepSeek-Coder-V2-Lite-Instruct | To factor the quadratic expression \(-12 x^2 + 411 x - \frac{4815}{2}\), we will follow these steps: 1. **Clear the fraction by multiplying the entire equation by 2**: \[ 2(-12 x^2 + 411 x - \frac{4815}{2}) = 2 \cdot -12 x^2 + 2 \cdot 411 x - 2 \cdot \frac{4815}{2} \] Simplifying this, we get: \[ -24 x^2 + 822 x - 4815 \] 2. **Factor out the greatest common divisor (GCD) of the coefficients**: The GCD of \(-24\), \(822\), and \(-4815\) is \(3\). So, we factor out \(3\): \[ -24 x^2 + 822 x - 4815 = 3(-8 x^2 + 274 x - 1605) \] 3. **Factor the quadratic inside the parentheses**: We need to factor \(-8 x^2 + 274 x - 1605\). To do this, we look for two numbers that multiply to \(-8 \times -1605 = 12840\) and add up to \(274\). The numbers that satisfy these conditions are \(30\) and \(428\) because: \[ 30 \times 428 = 12840 \quad \text{and} \quad 30 + 428 = 458 \] However, these numbers are incorrect. Let's re-evaluate the quadratic factorization. We should look for factors of \(-8 x^2 + 274 x - 1605\) that multiply to \(-8 \times -1605 = 12840\) and add up to \(274\). The correct factors are \(-15\) and \(107\) because: \[ -15 \times 107 = -1605 \quad \text{and} \quad -15 + 107 = 92 \] This is incorrect as well. Let's try another approach. 4. **Use the quadratic formula to find the roots and then factorize**: The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the quadratic \(-8 x^2 + 274 x - 1605\), we have \(a = -8\), \(b = 274\), and \(c = -1605\). Calculate the discriminant: \[ b^2 - 4ac = 274^2 - 4(-8)(-1605) = 75076 - 51360 = 23716 \] Calculate the roots: \[ x = \frac{-274 \pm \sqrt{23716}}{2(-8)} = \frac{-274 \pm 154}{-16} \] Calculate the two roots: \[ x_1 = \frac{-274 + 154}{-16} = \frac{-120}{-16} = 7.5 \] \[ x_2 = \frac{-274 - 154}{-16} = \frac{-428}{-16} = 26.75 \] These roots are not integers. Let's try another approach. 5. **Check the factorization again**: The correct factorization should be: \[ -12 x^2 + 411 x - \frac{4815}{2} = -12 \left(x - \frac{35}{2}\right) \left(x - \frac{43}{2}\right) \] Simplifying this: \[ -12 \left(2x - 35\right) \left(2x - 43\right) \] Therefore, the final answer is: \[ \boxed{-12 \left(2x - 35\right) \left(2x - 43\right)} \] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: To factor the given quadratic, we first need to identify the coefficients of the quadratic equation. The given quadratic is $-12x^2 + 411x - \frac{4815}{2}$. The coefficients are $a = -12$, $b = 411$, and $c = -\frac{4815}{2}$. ## Step 2: Next, we need to find two numbers whose product is $ac$ and whose sum is $b$. The product $ac$ is $-12 \times -\frac{4815}{2} = 28890$. We need to find two numbers whose product is $28890$ and whose sum is $411$. ## Step 3: To simplify the process, we can divide the product $ac$ by the coefficient of $x^2$, which is $-12$, to reduce the numbers we are working with. However, considering the complexity of the numbers involved, it might be more efficient to look for rational roots using the Rational Root Theorem or to apply the Quadratic Formula directly. ## Step 4: Since factoring the quadratic directly seems challenging due to the large numbers involved, we can apply the Quadratic Formula to find the roots of the equation. The Quadratic Formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Substituting the values of $a$, $b$, and $c$ into the formula will give us the roots of the equation. ## Step 5: Substitute the values of $a$, $b$, and $c$ into the Quadratic Formula. \[x = \frac{-411 \pm \sqrt{(411)^2 - 4(-12)(-\frac{4815}{2})}}{2(-12)}\] ## Step 6: Simplify the expression under the square root. \[x = \frac{-411 \pm \sqrt{168921 - 115560}}{-24}\] \[x = \frac{-411 \pm \sqrt{53361}}{-24}\] \[x = \frac{-411 \pm 231}{-24}\] ## Step 7: Solve for the two values of $x$. \[x_1 = \frac{-411 + 231}{-24} = \frac{-180}{-24} = \frac{15}{2}\] \[x_2 = \frac{-411 - 231}{-24} = \frac{-642}{-24} = \frac{107}{4}\] ## Step 8: Now that we have the roots, we can express the quadratic in factored form. Given the roots $x = \frac{15}{2}$ and $x = \frac{107}{4}$, the factored form of the quadratic is $-12(x - \frac{15}{2})(x - \frac{107}{4})$. ## Step 9: To finalize, simplify the expression if necessary. No further simplification is needed; the expression is already in its simplest factored form. The final answer is: $\boxed{-12(x - \frac{15}{2})(x - \frac{107}{4})}$ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $-12 x^2+411 x-\frac{4815}{2}$. ## Step 2: Determine the coefficients of the quadratic equation The coefficients of the quadratic equation are $a = -12$, $b = 411$, and $c = -\frac{4815}{2}$. ## Step 3: Use the quadratic formula to find the roots of the equation The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Plugging in the values of $a$, $b$, and $c$, we get $x = \frac{-411 \pm \sqrt{411^2 - 4(-12)(-\frac{4815}{2})}}{2(-12)}$. ## Step 4: Simplify the expression under the square root Simplifying the expression under the square root, we get $x = \frac{-411 \pm \sqrt{168921 - 115380}}{-24}$. ## Step 5: Further simplify the expression under the square root Further simplifying the expression under the square root, we get $x = \frac{-411 \pm \sqrt{53541}}{-24}$. ## Step 6: Simplify the square root Simplifying the square root, we get $x = \frac{-411 \pm 231}{-24}$. ## Step 7: Solve for the two possible values of x Solving for the two possible values of x, we get $x = \frac{-411 + 231}{-24}$ or $x = \frac{-411 - 231}{-24}$. ## Step 8: Simplify the expressions for x Simplifying the expressions for x, we get $x = \frac{-180}{-24}$ or $x = \frac{-642}{-24}$. ## Step 9: Further simplify the expressions for x Further simplifying the expressions for x, we get $x = \frac{15}{2}$ or $x = \frac{107}{4}$. ## Step 10: Write the factored form of the quadratic equation The factored form of the quadratic equation is $-12(x - \frac{15}{2})(x - \frac{107}{4})$. ## Step 11: Simplify the factored form Simplifying the factored form, we get $-12(x - \frac{15}{2})(x - \frac{107}{4}) = -12(\frac{2x - 15}{2})(\frac{4x - 107}{4}) = -3(2x - 15)(4x - 107)$. ## Step 12: Write the final factored form The final factored form is $-3(2x - 15)(4x - 107)$. The final answer is: $\boxed{-3(2x - 15)(4x - 107)}$ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| Qwen2-72B-Instruct | To factor the given quadratic expression \(-12x^2 + 411x - \frac{4815}{2}\), we can follow these steps: 1. **Find the discriminant** to check if it can be factored into real numbers. 2. **Use the quadratic formula** to find the roots, which will help us factor the expression. The general form of a quadratic equation is \(ax^2 + bx + c\). For our equation, \(a = -12\), \(b = 411\), and \(c = -\frac{4815}{2}\). ### Step 1: Find the Discriminant The discriminant is given by \(D = b^2 - 4ac\). \[D = (411)^2 - 4(-12)\left(-\frac{4815}{2}\right) = 169281 - 4(12)(2407.5) = 169281 - 115560 = 53721\] Since \(D > 0\), the quadratic has two distinct real roots, so it can be factored into real numbers. ### Step 2: Use the Quadratic Formula The roots of the quadratic equation are given by: \[x = \frac{-b \pm \sqrt{D}}{2a}\] Substituting the values: \[x = \frac{-411 \pm \sqrt{53721}}{2(-12)} = \frac{-411 \pm 232}{-24}\] This gives us two roots: \[x_1 = \frac{-411 + 232}{-24} = \frac{-179}{-24} = \frac{179}{24}\] \[x_2 = \frac{-411 - 232}{-24} = \frac{-643}{-24} = \frac{643}{24}\] ### Step 3: Factor the Quadratic A quadratic equation can be factored as \(a(x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) are the roots. So, the factored form of the given quadratic is: \[-12\left(x - \frac{179}{24}\right)\left(x - \frac{643}{24}\right)\] Simplifying this further: \[-12\left(\frac{24x - 179}{24}\right)\left(\frac{24x - 643}{24}\right)\] \[= \frac{-12}{24^2}(24x - 179)(24x - 643)\] \[= \frac{-1}{24}(24x - 179)(24x - 643)\] Thus, the factored form of the given quadratic is: \[\boxed{\frac{-1}{24}(24x - 179)(24x - 643)}\] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| baichuan4 | To factor the quadratic equation $-12x^2 + 411x - \frac{4815}{2}$, we first need to find two numbers $a$ and $b$ such that their product is equal to the coefficient of $x^2$ times the constant term, and their sum is equal to the coefficient of $x$. In this case, we have: $$ ab = (-12) \times \left(-\frac{4815}{2}\right) = 28890 $$ and $$ a + b = 411 $$ We need to find two numbers $a$ and $b$ that satisfy both of these equations. To do this, we can try to factor the product $ab$ into two factors whose sum is $411$. This may take some trial and error, but we can start by looking for factors of $28890$ that are close to each other, since their sum should be around $411$. One way to find these factors is to divide $28890$ by different numbers and see if the quotient is close to half of $411$. We can also use the divisors of $28890$ to help us find the pair. After some searching, we find that $360$ and $80$ are two numbers that multiply to $28800$, which is very close to $28890$. However, $360 + 80 = 440$ is not equal to $411$. We can adjust these numbers slightly to get a product of exactly $28890$ and a sum of exactly $411$. We find that $369$ and $80$ work, because: $$ 369 \times 80 = 29520 $$ and $$ 369 + 80 = 449 $$ This is still not quite right, so we continue to search for the correct pair. Eventually, we find that $360$ and $81$ are the correct numbers, because: $$ 360 \times 81 = 29160 $$ and $$ 360 + 81 = 441 $$ Now we have the correct pair of numbers, but we need to adjust them to match the coefficients of our original equation. Since our original equation has a coefficient of $-12$ on the $x^2$ term, we need to divide our $a$ and $b$ values by $-12$ to make them match. So, we have: $$ a = \frac{360}{-12} = -30 $$ and $$ b = \frac{81}{-12} = -\frac{81}{12} = -\frac{27}{4} $$ Now we can factor the quadratic equation as follows: $$ -12x^2 + 411x - \frac{4815}{2} = -12\left(x^2 - \frac{411}{-12}x + \frac{4815}{2 \cdot -12}\right) $$ $$ = -12\left(x^2 + \frac{411}{12}x - \frac{4815}{24}\right) $$ $$ = -12\left(x^2 + 34.25x - 200.625\right) $$ $$ = -12\left(x^2 + 30x + \frac{27}{4}x - 200.625\right) $$ $$ = -12\left((x + 30)(x + \frac{27}{4})\right) $$ Finally, we can simplify the expression by multiplying the $-12$ with the fractions inside the parentheses: $$ -12\left((x + 30)(x + \frac{27}{4})\right) = (-12x - 360)(x + \frac{27}{4}) $$ $$ = (-12x - 360)(4x + 27) $$ Since we want the leading coefficient of the quadratic to be positive, we can multiply the entire expression by $-1$ to get: $$ (12x + 360)(4x + 27) $$ This is the factored form of the quadratic equation. However, we need to check if this is equivalent to the original equation. We do this by expanding the factored form and comparing it to the original equation: $$ (12x + 360)(4x + 27) = 48x^2 + 324x + 1440x + 9720 $$ $$ = 48x^2 + 1764x + 9720 $$ This does not match the original equation, so we made a mistake somewhere. Let's go back and fix it. We need to find the correct factors of $28890$ that add up to $411$. We can use a factor tree or a calculator to help us find these factors. After some searching, we find that $360$ and $81$ are the correct numbers, because: $$ 360 \times 81 = 29160 $$ and $$ 360 + 81 = 441 $$ These numbers are close, but not exact. We need to find the exact pair of numbers that multiply to $28890$ and add up to $411$. These numbers are $360$ and $81$, because: $$ 360 \times 81 = 29160 $$ and $$ 360 + 81 = 441 $$ Now we have the correct pair of numbers, and we can factor the quadratic equation as follows: $$ -12x^2 + 411x - \frac{4815}{2} = -12\left(x^2 - \frac{411}{12}x + \frac{4815}{24}\right) $$ $$ = -12\left(x^2 - 34.25x + 200.625\right) $$ $$ = -12\left(x^2 - 30x - \frac{27}{4}x + 200.625\right) $$ $$ = -12\left((x - 30)(x - \frac{27}{4})\right) $$ Finally, we can simplify the expression by multiplying the $-12$ with the fractions inside the parentheses: $$ -12\left((x - 30)(x - \frac{27}{4})\right) = (-12x + 360)(x - \frac{27}{4}) $$ $$ = (-12x + 360)(4x - 27) $$ Since we want the leading coefficient of the quadratic to be positive, we can multiply the entire expression by $-1$ to get: $$ (12x - 360)(4x - 27) $$ This is the factored form of the quadratic equation. We can check if this is equivalent to the original equation by expanding the factored form and comparing it to the original equation: $$ (12x - 360)(4x - 27) = 48x^2 - 324x - 1440x + 9720 $$ $$ = 48x^2 - 1764x + 9720 $$ This matches the original equation, so we have the correct factored form. Our final answer is: $$ \boxed{(12x - 360)(4x - 27)} $$ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| doubao | For the quadratic $-12x^2 + 411x - \frac{4815}{2}$, we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = -12$, $b = 411$, $c = -\frac{4815}{2}$. The discriminant is $411^2 - 4 \times (-12) \times (-\frac{4815}{2}) = 168921 - 9630 \times 12 = 168921 - 115560 = 53361$. So $x = \frac{-411 \pm \sqrt{53361}}{-24} = \frac{-411 \pm 231}{-24}$. Taking the positive root: $x_1 = \frac{-411 + 231}{-24} = \frac{-180}{-24} = \frac{15}{2}$. Taking the negative root: $x_2 = \frac{-411 - 231}{-24} = \frac{-642}{-24} = \frac{107}{4}$. So the factored form is $-12(x - \frac{15}{2})(x - \frac{107}{4})$. Putting it in the box: $\boxed{-12(x - \frac{15}{2})(x - \frac{107}{4})}$. |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| gpto1 | To factor the quadratic expression \( -12x^2 + 411x - \dfrac{4815}{2} \), we start by eliminating the fraction: Multiply both sides by 2: \[ -24x^2 + 822x - 4815 = 0 \] Factor out -1 to make factorization simpler: \[ -1 \left( 24x^2 - 822x + 4815 \right) \] Next, we notice that all the coefficients are divisible by 3: \[ -1 \cdot 3 \left( 8x^2 - 274x + 1605 \right) \] Now, we factor the quadratic \( 8x^2 - 274x + 1605 \). Set up two binomials: \[ (8x - m)(x - n) \] We need \( m \) and \( n \) such that: \[ \begin{cases} mn = 1605 \\ 8n + m = 274 \end{cases} \] Substituting \( m = 1605/n \) into the second equation: \[ 8n + \frac{1605}{n} = 274 \] Multiply both sides by \( n \): \[ 8n^2 + 1605 = 274n \] Bring all terms to one side: \[ 8n^2 - 274n + 1605 = 0 \] Solve for \( n \) using the quadratic formula: \[ n = \frac{274 \pm \sqrt{274^2 - 4 \cdot 8 \cdot 1605}}{2 \cdot 8} \] \[ n = \frac{274 \pm 154}{16} \] We get \( n = 26.75 \) or \( n = 7.5 \) Using \( n = 7.5 \), we find \( m = 214 \) (since \( m = 1605/n \)). So the factors are: \[ (8x - 214)(x - 7.5) \] But since we prefer integer coefficients, we rewrite \( x - 7.5 = x - \dfrac{15}{2} \) and adjust the factors: \[ \begin{align*} (8x - 214)(x - 7.5) &= 2 \left( 4x - 107 \right) \left( x - \dfrac{15}{2} \right) \\ &= \left( 4x - 107 \right) \left( 2x - 15 \right) \end{align*} \] Finally, the original quadratic factors as: \[ -3 \left( 4x - 107 \right) \left( 2x - 15 \right) \] Therefore, the factorization of the original expression is: \(\boxed{-3\left(2x-15\right)\left(4x-107\right)}\) |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| kimi | To factor the quadratic expression \(-12x^2 + 411x - \frac{4815}{2}\), we will follow these steps: 1. **Rewrite the quadratic in standard form:** \[ -12x^2 + 411x - \frac{4815}{2} \] 2. **Factor out the greatest common divisor (GCD):** The GCD of the coefficients \(-12\), \(411\), and \(-\frac{4815}{2}\) is \(3\). Factor out \(3\) from each term: \[ -12x^2 + 411x - \frac{4815}{2} = 3(-4x^2 + 137x - \frac{1605}{2}) \] 3. **Simplify the expression inside the parentheses:** \[ -4x^2 + 137x - \frac{1605}{2} \] 4. **Rewrite the quadratic inside the parentheses in standard form:** \[ -4x^2 + 137x - 802.5 \] 5. **Factor the quadratic inside the parentheses:** We need to find two numbers that multiply to \(-4 \times (-802.5) = 3210\) and add to \(137\). These numbers are \(90\) and \(35\). Rewrite the middle term using these numbers: \[ -4x^2 + 90x - 35x - 802.5 \] 6. **Group the terms and factor by grouping:** \[ (-4x^2 + 90x) - (35x + 802.5) \] Factor out the common factors from each group: \[ 4x(-x + 22.5) - 35(-x + 22.5) \] 7. **Factor out the common binomial factor \((-x + 22.5)\):** \[ (4x - 35)(-x + 22.5) \] 8. **Combine the factored form with the GCD factored out earlier:** \[ 3(4x - 35)(-x + 22.5) \] 9. **Simplify the expression by distributing the \(-1\) in the second factor:** \[ 3(4x - 35)(x - 22.5) \] Thus, the factored form of the quadratic expression is: \[ \boxed{3(4x - 35)(x - 22.5)} \] |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| minimax | To factor the quadratic $-12 x^2 + 411 x - \frac{4815}{2}$, we need to find two numbers that multiply to give the product of the coefficient of $x^2$ (which is $-12$) and the constant term (which is $-\frac{4815}{2}$), and at the same time, these two numbers should add up to the coefficient of $x$ (which is $411$). First, let's find the product of the coefficient of $x^2$ and the constant term: Product = $(-12) \cdot \left(-\frac{4815}{2}\right) = 12 \cdot \frac{4815}{2} = 6 \cdot 4815 = 28890$ Now we need to find two numbers that multiply to $28890$ and add up to $411$. Since the numbers are large, we can use trial and error or a systematic approach to find the factors of $28890$ that add up to $411$. Let's start by factoring $ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| wenxinyiyan | To factor the given quadratic, we can use a method called "completing the square" or find two numbers that multiply to the constant term and add to the coefficient of the linear term. However, in this case, the quadratic is not easily factorable using integer coefficients, so we will use the quadratic formula to find the roots and then express the quadratic as a product of its roots. The quadratic formula is: $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$ Applied to the given quadratic $-12x^2 + 411x - \frac{4815}{2}$, we have: $a = -12, \quad b = 411, \quad c = -\frac{4815}{2}$ First, we calculate the discriminant: $\Delta = b^2 - 4ac = 411^2 - 4(-12)(-\frac{4815}{2}) = 168921 - 115560 = 53361$ Now, we find the roots: $x_1 = \frac{{-411 + \sqrt{{53361}}}}{-24} = \frac{{-411 + 231}}{-24} = \frac{{-180}}{-24} = \frac{15}{2}$ $x_2 = \frac{{-411 - \sqrt{{53361}}}}{-24} = \frac{{-411 - 231}}{-24} = \frac{{-642}}{-24} = \frac{107}{4}$ Therefore, the quadratic can be factored as: $-12x^2 + 411x - \frac{4815}{2} = -12(x - \frac{15}{2})(x - \frac{107}{4})$ However, the roots found above do not exactly match the original quadratic due to rounding errors in the calculation of the discriminant and the roots. Let's recalculate more precisely: $\Delta = 411^2 - 4(-12)(-\frac{4815}{2}) = 168921 - 115560 = 53361$ $x_1 = \frac{{-411 + \sqrt{{53361}}}}{-24} = \frac{{-411 + 231.376}}{-24} = \frac{{-179.624}}{-24} = 7.484333$ $x_2 = \frac{{-411 - \sqrt{{53361}}}}{-24} = \frac{{-411 - 231.376}}{-24} = \frac{{-642.376}}{-24} = 26.765667$ So, the precise factored form is: $-12x^2 + 411x - \frac{4815}{2} = -12(x - 7.484333)(x - 26.765667)$ However, since the original request was to put the final answer in a box, and the answer should be finite and exact, we should recognize that the roots found are not exact and the quadratic cannot be exactly factored using rational numbers. Thus, we cannot provide an exact boxed answer as requested. Instead, we can express that the quadratic cannot be simply factored: $\boxed{-12x^2 + 411x - \frac{4815}{2} \text{ cannot be simply factored.}}$ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| zhipu | We can factor out a $-\frac{1}{2}$ from the quadratic to simplify it: $$-12 x^2 + 411 x - \frac{4815}{2} = -\frac{1}{2}(24 x^2 - 822 x + 4815).$$ Now we need to factor the quadratic inside the parentheses. We are looking for two numbers that multiply to $24 \cdot 4815$ and add up to $-822$. The prime factorization of $24$ is $2^3 \cdot 3$, and the prime factorization of $4815$ is $3 \cdot 5 \cdot 323$. We can see that $3$ is a common factor, so we can start by pairing $3$ with $1605$ (since $3 \cdot 1605 = 4815$). However, $1605$ is not a factor of $24$, so we need to find a pair of factors of $1605$ that, when multiplied by the appropriate factors of $24$, will add up to $-822$. $1605$ can be factored as $5 \cdot 323$, and $323$ can be factored as $17 \cdot 19$. Now we need to find a pair of factors from $2^3 \cdot 3$ and $5 \cdot 17 \cdot 19$ that add up to $-822$. After trying different combinations, we find that $-57$ and $-45$ work because: $$-57 \cdot -45 = 2565,$$ and $$-57 + (-45) = -102.$$ Since we need a sum of $-822$, we need to multiply these numbers by $8$ (a factor of $24$) to get the correct sum: $$-57 \cdot 8 = -456,$$ $$-45 \cdot 8 = -360,$$ and $$-456 + (-360) = -816.$$ We are close, but we need a sum of $-822$, not $-816$. This means we need to adjust our factors slightly. Let's try $-59$ and $-43$: $$-59 \cdot -43 = 2537,$$ and $$-59 + (-43) = -102.$$ Again, we multiply by $8$ to get closer to $-822$: $$-59 \cdot 8 = -472,$$ $$-43 \cdot 8 = -344,$$ and $$-472 + (-344) = -816.$$ We are still short by $6$. This means we need to adjust our factors again. Let's try $-61$ and $-41$: $$-61 \cdot -41 = 2501,$$ and $$-61 + (-41) = -102.$$ Multiplying by $8$: $$-61 \cdot 8 = -488,$$ $$-41 \cdot 8 = -328,$$ and $$-488 + (-328) = -816.$$ We are still short by $6$. Let's try one more adjustment: $-60$ and $-42$: $$-60 \cdot -42 = 2520,$$ and $$-60 + (-42) = -102.$$ Multiplying by $8$: $$-60 \cdot 8 = -480,$$ $$-42 \cdot 8 = -336,$$ and $$-480 + (-336) = -816.$$ We are still short by $6$. It seems we have made a mistake in our approach. Let's go back and try a different combination of factors. We need to find two numbers that multiply to $24 \cdot 4815$ and add up to $-822$. Let's try $-81$ and $-41$: $$-81 \cdot -41 = 3321,$$ and $$-81 + (-41) = -122.$$ Multiplying by $6$ (a factor of $24$): $$-81 \cdot 6 = -486,$$ $$-41 \cdot 6 = -246,$$ and $$-486 + (-246) = -732.$$ We are still not at $-822$. Let's try $-57$ and $-45$ again, but this time let's multiply by $9$ (another factor of $24$): $$-57 \cdot 9 = -513,$$ $$-45 \cdot 9 = -405,$$ and $$-513 + (-405) = -918.$$ This is too much. We need to find the correct combination of factors. Let's try $-59$ and $-43$ again, but this time let's multiply by $9$: $$-59 \cdot 9 = -531,$$ $$-43 \cdot 9 = -387,$$ and $$-531 + (-387) = -918.$$ This is still too much. We need to find a combination that gives us $-822$ when multiplied by a factor of $24$. Let's try $-61$ and $-41$ again, but this time let's multiply by $9$: $$-61 \cdot 9 = -549,$$ $$-41 \cdot 9 = -369,$$ and $$-549 + (-369) = -918.$$ This is still too much. We need to find a combination that gives us $-822$ when multiplied by a factor of $24$. Let's try $-60$ and $-42$ again, but this time let's multiply by $9$: $$-60 \cdot 9 = -540,$$ $$-42 \cdot 9 = -378,$$ and $$-540 + (-378) = -918.$$ This is still too much. We need to find a combination that gives us $-822$ when multiplied by a factor of $24$. Let's try $-58$ and $-44$: $$-58 \cdot -44 = 2552,$$ and $$-58 + (-44) = -102.$$ Multiplying by $8$: $$-58 \cdot 8 = -464,$$ $$-44 \cdot 8 = -352,$$ and $$-464 + (-352) = -816.$$ We are still short by $6$. Let's try $-57$ and $-45$ again, but this time let's multiply by $10$: $$-57 \cdot 10 = -570,$$ $$-45 \cdot 10 = -450,$$ and $$-570 + (-450) = -1020.$$ This is too much. Let's try $-59$ and $-43$ again, but this time let's multiply by $10$: $$-59 \cdot 10 = -590,$$ $$-43 \cdot 10 = -430,$$ and $$-590 + (-430) = -1020.$$ This is too much. Let's try $-61$ and $-41$ again, but this time let's multiply by $10$: $$-61 \cdot 10 = -610,$$ $$-41 \cdot 10 = -410,$$ and $$-610 + (-410) = -1020.$$ This is too much. Let's try $-60$ and $-42$ again, but this time let's multiply by $10$: $$-60 \cdot 10 = -600,$$ $$-42 \cdot 10 = -420,$$ and $$-600 + (-420) = -1020.$$ This is too much. Let's try $-58$ and $-44$ again, but this time let's multiply by $10$: $$-58 \cdot 10 = -580,$$ $$-44 \cdot 10 = -440,$$ and $$-580 + (-440) = -1020.$$ This is too much. We need to find a combination that gives us $-822$ when multiplied by a factor of $24$. Let's try $-57$ and $-45$ again, but this time let's multiply by $9$: $$-57 \cdot 9 = -513,$$ $$-45 \cdot 9 = -405,$$ and $$-513 + (-405) = -918.$$ This is too much. Let's try $-59$ and $-43$ again, but this time let's multiply by $9$: $$-59 \cdot 9 = -531,$$ $$-43 \cdot 9 = -387,$$ and $$-531 + (-387) = -918.$$ This is too much. Let's try $-61$ and $-41$ again, but this time let's multiply by $9$: $$-61 \cdot 9 = -549,$$ $$-41 \cdot 9 = -369,$$ and $$-549 + (-369) = -918.$$ This is too much. Let's try $-60$ and $-42$ again, but this time let's multiply by $9$: $$-60 \cdot 9 = -540,$$ $$-42 \cdot 9 = -378,$$ and $$-540 + (-378) = -918.$$ This is too much. Let's try $-58$ and $-44$ again, but this time let's multiply by $9$: $$-58 \cdot 9 = -522,$$ $$-44 \cdot 9 = -396,$$ and $$-522 + (-396) = -918.$$ This is too much. Let's try $-57$ and $-45$ again, but this time let's multiply by $8$: $$-57 \cdot 8 = -456,$$ $$-45 \cdot 8 = -360,$$ and $$-456 + (-360) = -816.$$ We are still short by $6$. Let's try $-59$ and $-43$ again, but this time let's multiply by $8$: $$-59 \cdot 8 = -472,$$ $$-43 \cdot 8 = -344,$$ and $$-472 + (-344) = -816.$$ We are still short by $6$. Let's try $-61$ and $-41$ again, but this time let's multiply by $8$: $$-61 \cdot 8 = -488,$$ $$-41 \cdot 8 = -328,$$ and $$-488 + (-328) = -816.$$ We are still short by $6$. Let's try $-60$ and $-42$ again, but this time let's multiply by $8$: $$-60 \cdot 8 = -480,$$ $$-42 \cdot 8 = -336,$$ and $$-480 + (-336) = -816.$$ We are still short by $6$. Let's try $-58$ and $-44$ again, but this time let's multiply by $8$: $$-58 \cdot 8 = -464,$$ $$-44 \cdot 8 = -352,$$ and $$-464 + (-352) = -816.$$ We are still short by $6$. Let's try $-57$ and $-45$ again, but this time let's multiply by $7$: $$-57 \cdot 7 = -399,$$ $$-45 \cdot 7 = -315,$$ and $$-399 + (-315) = -714.$$ This is too little. Let's try $-59$ and $-43$ again, but this time let's multiply by $7$: $$-59 \cdot 7 = -413,$$ $$-43 \cdot 7 = -301,$$ and $$-413 + (-301) = -714.$$ This is too little. Let's try $-61$ and $-41$ again, but this time let's multiply by $7$: $$-61 \cdot 7 = -427,$$ $$-41 \cdot 7 = -287,$$ and $$-427 + (-287) = -714.$$ This is too little. Let's try $-60$ and $-42$ again, but this time let's multiply by $7$: $$-60 \cdot 7 = -420,$$ $$-42 \cdot 7 = -294,$$ and $$-420 + (-294) = -714.$$ This is too little. Let's try $-58$ and $-44$ again, but this time let's multiply by $7$: $$-58 \cdot 7 = -406,$$ $$-44 \cdot 7 = -308,$$ and $$-406 + (-308) = -714.$$ This is too little. Let's try $-57$ and $-45$ again, but this time let's multiply by $6$: $$-57 \cdot 6 = -342,$$ $$-45 \cdot 6 = -270,$$ and $$-342 + (-270) = -612.$$ This is too little. Let's try $-59$ and $-43$ again, but this time let's multiply by $6$: $$-59 \cdot 6 = -354,$$ $$-43 \cdot 6 = -258,$$ and $$-354 + (-258) = -612.$$ This is too little. Let's try $-61$ and $-41$ again, but this time let's multiply by $6$: $$-61 \cdot 6 = -366,$$ $$-41 \cdot 6 = -246,$$ and $$-366 + (-246) = -612.$$ This is too little. Let's try $-60$ and $-42$ again, but this time let's multiply by $6$: $$-60 \cdot 6 = -360,$$ $$-42 \cdot 6 = -252,$$ and $$-360 + (-252) = -612.$$ This is too little. Let's try $-58$ and $-44$ again, but this time let's multiply by $6$: $$-58 \cdot 6 = -348,$$ $$-44 \cdot 6 = -264,$$ and $$-348 + (-264) = -612.$$ This is too little. Let's try $-57$ and $-45$ again, but this time let's multiply by $5$: $$-57 \cdot 5 = -285,$$ $$-45 \cdot 5 = -225,$$ and $$-285 + (-225) = -510.$$ This is too little. Let's try $-59$ and $-43$ again, but this time let's multiply by $5$: $$-59 \cdot 5 = -295,$$ $$-43 \cdot 5 = -215,$$ and $$-295 + (-215) = -510.$$ This is too little. Let's try $-61$ and $-41$ again, but this time let's multiply by $5$: $$-61 \cdot 5 = -305,$$ $$-41 \cdot 5 = -205,$$ and $$-305 + (-205) = -510.$$ This is too little. Let's try $-60$ and $-42$ again, but this time let's multiply by $5$: $$-60 \cdot 5 = -300,$$ $$-42 \cdot 5 = -210,$$ and $$-300 + (-210) = -510.$$ This is too little. Let's try $-58$ and $-44$ again, but this time let's multiply by $5$: $$-58 \cdot 5 = -290,$$ $$-44 \cdot 5 = -220,$$ and $$-290 + (-220) = -510.$$ This is too little. Let's try $-57$ and $-45$ again, but this time let's multiply by $4$: $$-57 \cdot 4 = -228,$$ $$-45 \cdot 4 = -180,$$ and $$-228 + (-180) = -408.$$ This is too little. Let's try $-59$ and $-43$ again, but this time let's multiply by $4$: $$-59 \cdot 4 = -236,$$ $$-43 \cdot 4 = -172,$$ and $$-236 + (-172) = -408.$$ This is too little. Let's try $-61$ and $-41$ again, but this time let's multiply by $4$: $$-61 \cdot 4 = -244,$$ $$-41 \cdot 4 = -164,$$ and $$-244 + (-164) = -408.$$ This is too little. Let's try $-60$ and $-42$ again, but this time let's multiply by $4$: $$-60 \cdot 4 = -240,$$ $$-42 \cdot 4 = -168,$$ and $$-240 + (-168) = -408.$$ This is too little. Let's try $-58$ and $-44$ again, but this time let's multiply by $4$: $$-58 \cdot 4 = -232,$$ $$-44 \cdot 4 = -176,$$ and $$-232 + (-176) = -408.$$ This is too little. Let's try $-57$ and $-45$ again, but this time let's multiply by $3$: $$-57 \cdot 3 = -171,$$ $$-45 \cdot 3 = -135,$$ and $$-171 + (-135) = -306.$$ This is too little. Let's try $-59$ and $-43$ |
12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-1833a45c-ef66-9573-9f8c-9f1558a8ac2b', 'request_id': '1833a45c-ef66-9573-9f8c-9f1558a8ac2b'} | 12 \left(\frac{107}{4}-x\right) \left(x-\frac{15}{2}\right) |