| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To factor the quadratic expression \(10 x^2 + \frac{260 x}{\sqrt{3}} - 1050\), we will follow these steps: 1. **Factor out the common factor**: Notice that each term in the quadratic expression has a common factor of 10. We can factor out 10 from each term: \[ 10 x^2 + \frac{260 x}{\sqrt{3}} - 1050 = 10 \left( x^2 + \frac{26 x}{\sqrt{3}} - 105 \right) \] 2. **Simplify the expression inside the parentheses**: We need to simplify the quadratic expression inside the parentheses: \[ x^2 + \frac{26 x}{\sqrt{3}} - 105 \] 3. **Rationalize the coefficient of \(x\)**: To make the coefficient of \(x\) a rational number, we multiply the entire quadratic expression by \(\sqrt{3}\): \[ \sqrt{3} \left( x^2 + \frac{26 x}{\sqrt{3}} - 105 \right) = \sqrt{3} x^2 + 26 x - 105 \sqrt{3} \] 4. **Factor the simplified quadratic expression**: Now, we need to factor the quadratic expression \(\sqrt{3} x^2 + 26 x - 105 \sqrt{3}\). We look for two numbers that multiply to \(-105 \sqrt{3}\) and add to 26. These numbers are 35 and -3: \[ \sqrt{3} x^2 + 26 x - 105 \sqrt{3} = (\sqrt{3} x + 35)(x - 3) \] 5. **Combine the factors**: Recall that we initially factored out 10. Therefore, we need to include this factor in our final answer: \[ 10 \left( \sqrt{3} x + 35 \right) \left( x - 3 \right) \] Thus, the factored form of the quadratic expression is: \[ \boxed{10 (\sqrt{3} x + 35)(x - 3)} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| deepseek-r1 | To factor the quadratic \(10 x^2 + \frac{260 x}{\sqrt{3}} - 1050\), we start by rationalizing the denominator of the middle term: \[ \frac{260 x}{\sqrt{3}} = \frac{260 x \sqrt{3}}{3} \] Rewriting the quadratic with this term: \[ 10 x^2 + \frac{260 \sqrt{3}}{3} x - 1050 \] Next, we factor out the common factor of 10: \[ 10 \left( x^2 + \frac{26 \sqrt{3}}{3} x - 105 \right) \] We use the substitution \(y = x \sqrt{3}\), which transforms the quadratic inside the brackets. Substituting \(x = \frac{y}{\sqrt{3}}\) into the quadratic expression: \[ \left( \frac{y}{\sqrt{3}} \right)^2 + \frac{26 \sqrt{3}}{3} \left( \frac{y}{\sqrt{3}} \right) - 105 \] Simplifying each term: \[ \frac{y^2}{3} + \frac{26 y}{3} - 105 \] Factoring out \(\frac{1}{3}\): \[ \frac{1}{3} \left( y^2 + 26 y - 315 \right) \] We factor the quadratic \(y^2 + 26 y - 315\) by finding two numbers that multiply to \(-315\) and add to \(26\). These numbers are \(35\) and \(-9\): \[ (y + 35)(y - 9) \] Substituting back \(y = x \sqrt{3}\): \[ \frac{1}{3} (x \sqrt{3} + 35)(x \sqrt{3} - 9) \] Multiplying by the factored-out 10: \[ 10 \cdot \frac{1}{3} (x \sqrt{3} + 35)(x \sqrt{3} - 9) \] Simplifying the expression: \[ \frac{10}{3} (x \sqrt{3} + 35)(x \sqrt{3} - 9) \] Rationalizing the terms inside the factors: \[ x \sqrt{3} + 35 = \sqrt{3} \left( x + \frac{35 \sqrt{3}}{3} \right) \] \[ x \sqrt{3} - 9 = \sqrt{3} \left( x - 3 \sqrt{3} \right) \] Thus, the factored form is: \[ 10 \left( x - 3 \sqrt{3} \right) \left( x + \frac{35 \sqrt{3}}{3} \right) \] The final answer is: \[ \boxed{10 \left( x - 3\sqrt{3} \right) \left( x + \frac{35\sqrt{3}}{3} \right)} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| gpto1 | To factor the quadratic expression \(10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050\), we'll aim to break it down into its component factors. Here's a step-by-step approach: **Step 1: Simplify the expression by eliminating radicals in denominators.** First, to eliminate denominators, multiply both sides of the equation by \(\sqrt{3}\): \[ \sqrt{3} \times \left(10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050\right) = 0. \] Simplify: \[ 10 \sqrt{3} x^2 + 260 x - 1050 \sqrt{3} = 0. \] **Step 2: Factor the simplified quadratic expression.** We can rewrite the quadratic as: \[ 10 \sqrt{3} x^2 + 260 x - 1050 \sqrt{3} = 10 \left( \sqrt{3} x^2 + 26 x - 105 \sqrt{3} \right). \] Next, factor the quadratic inside the parentheses. To do this, consider factoring it as \((\sqrt{3} x + a)(x + b)\). Expanding the product, we get: \[ \sqrt{3} x^2 + (\sqrt{3} b + a)x + a b = \sqrt{3} x^2 + 26 x - 105 \sqrt{3}. \] Set up the system of equations: \[ \begin{cases} \sqrt{3} b + a = 26,\\ a b = -105 \sqrt{3}. \end{cases} \] Let \(a = 26 - \sqrt{3} b\). Substitute back into \(a b\): \[ (26 - \sqrt{3} b) b = -105 \sqrt{3} \implies 26 b - \sqrt{3} b^2 = -105 \sqrt{3}. \] Multiply both sides by \(-1\) to make calculations easier: \[ \sqrt{3} b^2 - 26 b + 105 \sqrt{3} = 0. \] Let \(b = k \sqrt{3}\). Substituting and simplifying, we get the quadratic equation: \[ 3 k^2 - 26 k - 105 = 0. \] Solving this quadratic equation, we find \(k = \dfrac{35}{3}\) or \(k = -3\). **Step 3: Determine the factors.** Corresponding to \(k = \dfrac{35}{3}\): \[ b = \dfrac{35}{3} \sqrt{3}, \quad a = 26 - 3 k = -9. \] Corresponding to \(k = -3\): \[ b = -3 \sqrt{3}, \quad a = 26 - 3 (-3) = 35. \] Using the solution that results in integer coefficients in \(a\) and \(b\), we select \(k = -3\). Therefore, the factors are: \[ \sqrt{3} x^2 + 26 x - 105 \sqrt{3} = (\sqrt{3} x + 35)(x - 3 \sqrt{3}). \] **Step 4: Write the final factored form.** Including the factor of \(10\) we factored out earlier, the original quadratic expression factors completely as: \[ 10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050 = 10 (\sqrt{3} x + 35)(x - 3 \sqrt{3}). \] Therefore, the factored form is: \(\boxed{10\,(\sqrt{3}x+35)\left(x-3\sqrt{3}\right)}\) |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| gpt-4o-2024-08-06 | To factor the quadratic \(10x^2 + \frac{260x}{\sqrt{3}} - 1050\), we start by simplifying the expression. First, let's clear the fraction by multiplying the entire quadratic by \(\sqrt{3}\): \[ \sqrt{3} \cdot \left(10x^2 + \frac{260x}{\sqrt{3}} - 1050\right) = 10\sqrt{3}x^2 + 260x - 1050\sqrt{3} \] Now, we have the quadratic: \[ 10\sqrt{3}x^2 + 260x - 1050\sqrt{3} \] Next, we will factor this quadratic expression. We need to find two numbers that multiply to the product of the coefficient of \(x^2\) term and the constant term, and add up to the coefficient of the \(x\) term. The product is: \[ 10\sqrt{3} \times (-1050\sqrt{3}) = -10500 \times 3 = -31500 \] We need two numbers that multiply to \(-31500\) and add to \(260\). After testing several pairs, we find that \(450\) and \(-70\) work because: \[ 450 \times (-70) = -31500 \quad \text{and} \quad 450 + (-70) = 380 \] However, we need to find numbers that add to \(260\), not \(380\). Let's re-evaluate our approach. We need to find numbers that multiply to \(-31500\) and add to \(260\). Upon further inspection, the correct numbers are \(525\) and \(-60\) because: \[ 525 \times (-60) = -31500 \quad \text{and} \quad 525 + (-60) = 465 \] This is incorrect. Let's try another approach by using the quadratic formula to find the roots and then factor. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 10\sqrt{3}\), \(b = 260\), and \(c = -1050\sqrt{3}\). Calculate the discriminant: \[ b^2 - 4ac = 260^2 - 4 \times 10\sqrt{3} \times (-1050\sqrt{3}) \] \[ = 67600 + 4 \times 10 \times 1050 \times 3 \] \[ = 67600 + 126000 \] \[ = 193600 \] The discriminant is a perfect square, \(\sqrt{193600} = 440\). Now, find the roots: \[ x = \frac{-260 \pm 440}{20\sqrt{3}} \] Calculate the roots: 1. \(x = \frac{-260 + 440}{20\sqrt{3}} = \frac{180}{20\sqrt{3}} = \frac{9}{\sqrt{3}} = 3\sqrt{3}\) 2. \(x = \frac{-260 - 440}{20\sqrt{3}} = \frac{-700}{20\sqrt{3}} = \frac{-35}{\sqrt{3}} = -35\sqrt{3}\) Thus, the quadratic factors as: \[ 10\sqrt{3}(x - 3\sqrt{3})(x + 35\sqrt{3}) \] Finally, the factorization of the original quadratic is: \[ \boxed{(x - 3\sqrt{3})(x + 35\sqrt{3})} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| DeepSeek-Coder-V2-Lite-Instruct | To factor the quadratic expression \(10 x^2 + \frac{260 x}{\sqrt{3}} - 1050\), we will follow these steps: 1. **Factor out the greatest common factor (GCF) from the terms**: The GCF of the coefficients \(10\), \(\frac{260}{\sqrt{3}}\), and \(-1050\) is \(10\). \[ 10 x^2 + \frac{260 x}{\sqrt{3}} - 1050 = 10 \left( x^2 + \frac{26 x}{\sqrt{3}} - 105 \right) \] 2. **Simplify the expression inside the parentheses**: To simplify the expression inside the parentheses, we need to rationalize the denominator. However, it is often easier to factor the quadratic expression directly without rationalizing the denominator. 3. **Factor the quadratic expression \(x^2 + \frac{26 x}{\sqrt{3}} - 105\)**: We need to find two numbers that multiply to \(-105\) (the constant term) and add up to \(\frac{26}{\sqrt{3}}\) (the coefficient of the linear term). Let's denote the quadratic as \(x^2 + \frac{26 x}{\sqrt{3}} - 105\). To factor this, we look for two numbers that multiply to \(-105\) and add up to \(\frac{26}{\sqrt{3}}\). The numbers that satisfy these conditions are \(\frac{15}{\sqrt{3}}\) and \(-\frac{7}{\sqrt{3}}\) because: \[ \left( \frac{15}{\sqrt{3}} \right) \left( -\frac{7}{\sqrt{3}} \right) = -105 \] and \[ \frac{15}{\sqrt{3}} + \left( -\frac{7}{\sqrt{3}} \right) = \frac{15 - 7}{\sqrt{3}} = \frac{8}{\sqrt{3}} = \frac{26}{\sqrt{3}} \] 4. **Rewrite the quadratic expression using these numbers**: \[ x^2 + \frac{26 x}{\sqrt{3}} - 105 = \left( x + \frac{15}{\sqrt{3}} \right) \left( x - \frac{7}{\sqrt{3}} \right) \] 5. **Combine the factored terms with the GCF**: \[ 10 \left( x + \frac{15}{\sqrt{3}} \right) \left( x - \frac{7}{\sqrt{3}} \right) \] 6. **Simplify the terms inside the parentheses**: To simplify \(\frac{15}{\sqrt{3}}\) and \(\frac{7}{\sqrt{3}}\), we rationalize the denominators: \[ \frac{15}{\sqrt{3}} = \frac{15 \sqrt{3}}{3} = 5 \sqrt{3} \] \[ \frac{7}{\sqrt{3}} = \frac{7 \sqrt{3}}{3} = \frac{7 \sqrt{3}}{3} \] Therefore, the factored form becomes: \[ 10 \left( x + 5 \sqrt{3} \right) \left( x - \frac{7 \sqrt{3}}{3} \right) \] 7. **Final factored form**: \[ 10 \left( x + 5 \sqrt{3} \right) \left( x - \frac{7 \sqrt{3}}{3} \right) \] However, this form is not in the simplest factored form. Let's re-evaluate the factorization: \[ 10 \left( x + \frac{15}{\sqrt{3}} \right) \left( x - \frac{7}{\sqrt{3}} \right) \] Simplifying further: \[ 10 \left( x + 5 \sqrt{3} \right) \left( x - \frac{7 \sqrt{3}}{3} \right) \] This is the correct factored form. Therefore, the final answer is: \[ \boxed{10 \left( x + 5 \sqrt{3} \right) \left( x - \frac{7 \sqrt{3}}{3} \right)} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: To factor the given quadratic, we first need to simplify it by getting rid of the fraction and making the coefficient of $x^2$ equal to 1. We can achieve this by dividing the entire quadratic by 10 and then multiplying the $x$ term by $\sqrt{3}$ to rationalize the denominator. ## Step 2: After simplifying, the quadratic becomes $x^2+\frac{26}{\sqrt{3}}x-105$. Now, we need to find two numbers whose product is -105 and whose sum is $\frac{26}{\sqrt{3}}$. ## Step 3: To make things easier, let's multiply the entire quadratic by $\sqrt{3}$ to get rid of the fraction. This gives us $\sqrt{3}x^2+26x-105\sqrt{3}$. ## Step 4: Now, we need to find two numbers whose product is $-105\sqrt{3}$ and whose sum is 26. Let's try to factor the quadratic by finding these two numbers. ## Step 5: The factors of $-105\sqrt{3}$ are $-15\sqrt{3}$ and 7, or $-21\sqrt{3}$ and 5, or $-35\sqrt{3}$ and 3, or $-105\sqrt{3}$ and 1. We need to find the pair whose sum is 26. ## Step 6: After trying different pairs, we find that $-15\sqrt{3}$ and 7 do not add up to 26, $-21\sqrt{3}$ and 5 do not add up to 26, and $-35\sqrt{3}$ and 3 do not add up to 26. However, $-15\sqrt{3}$ and 7 are not the correct pair, $-21\sqrt{3}$ and 5 are not the correct pair, and $-35\sqrt{3}$ and 3 are not the correct pair. But $-15\sqrt{3}$ is not one of the factors of the quadratic. ## Step 7: Upon re-examining the factors, we find that $-15\sqrt{3}$ is not one of the factors, but 15 is a factor of 105, and $\sqrt{3}$ is already present in the quadratic. Let's try to factor the quadratic again, and we find that $-15\sqrt{3}$ is actually $-15$ times $\sqrt{3}$. ## Step 8: Now, we can rewrite the quadratic as $(\sqrt{3}x-15)(x+7)$. This is the factored form of the given quadratic. ## Step 9: However, we still need to multiply the entire quadratic by 10 to get the original quadratic. So, the final answer will be $10(\sqrt{3}x-15)(x+7)$. ## Step 10: We can now simplify the expression to get the final answer. $10(\sqrt{3}x-15)(x+7)$ is already simplified. The final answer is: $\boxed{10(\sqrt{3}x-15)(x+7)}$ |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $10 x^2+\frac{260 x}{\sqrt{3}}-1050$. ## Step 2: Determine the coefficients of the quadratic equation The coefficients of the quadratic equation are $a = 10$, $b = \frac{260}{\sqrt{3}}$, and $c = -1050$. ## 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 $x = \frac{-\frac{260}{\sqrt{3}} \pm \sqrt{\left(\frac{260}{\sqrt{3}}\right)^2 - 4(10)(-1050)}}{2(10)}$. ## Step 4: Simplify the expression under the square root Simplifying the expression under the square root, we get $\left(\frac{260}{\sqrt{3}}\right)^2 - 4(10)(-1050) = \frac{67600}{3} + 42000 = \frac{67600 + 126000}{3} = \frac{193600}{3}$. ## Step 5: Simplify the quadratic formula expression Simplifying the quadratic formula expression, we get $x = \frac{-\frac{260}{\sqrt{3}} \pm \sqrt{\frac{193600}{3}}}{20} = \frac{-\frac{260}{\sqrt{3}} \pm \frac{440}{\sqrt{3}}}{20}$. ## Step 6: Simplify the expression further Simplifying the expression further, we get $x = \frac{-260 \pm 440}{20\sqrt{3}} = \frac{-260 + 440}{20\sqrt{3}}$ or $x = \frac{-260 - 440}{20\sqrt{3}}$. ## Step 7: Simplify the two possible values of x Simplifying the two possible values of x, we get $x = \frac{180}{20\sqrt{3}} = \frac{9}{\sqrt{3}} = 3\sqrt{3}$ or $x = \frac{-700}{20\sqrt{3}} = \frac{-35}{\sqrt{3}} = -\frac{35\sqrt{3}}{3}$. ## Step 8: Write the factored form of the quadratic equation The factored form of the quadratic equation is $10(x - 3\sqrt{3})(x + \frac{35\sqrt{3}}{3})$. ## Step 9: Simplify the factored form Simplifying the factored form, we get $10(x - 3\sqrt{3})(x + \frac{35\sqrt{3}}{3}) = 10(x - 3\sqrt{3})(\frac{3x + 35\sqrt{3}}{3}) = \frac{10}{3}(3x - 9\sqrt{3})(x + \frac{35\sqrt{3}}{3}) = \frac{10}{3}(3x - 9\sqrt{3})(3x + 35\sqrt{3}) = \frac{10}{3}(9x^2 + 26x\sqrt{3} - 315)$. ## Step 10: Write the final factored form The final factored form is $\boxed{10(x - 3\sqrt{3})(x + \frac{35\sqrt{3}}{3})}$. The final answer is: $\boxed{10(x - 3\sqrt{3})(x + \frac{35\sqrt{3}}{3})}$ |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| Qwen2-72B-Instruct | To factor the given quadratic expression, we first simplify it by factoring out the greatest common factor (GCF). The GCF of \(10x^2\), \(\frac{260x}{\sqrt{3}}\), and \(-1050\) is \(10\). So, we can write: \[10x^2 + \frac{260x}{\sqrt{3}} - 1050 = 10(x^2 + \frac{26x}{\sqrt{3}} - 105)\] Next, to simplify further, let's rationalize the denominator of the second term: \[\frac{26x}{\sqrt{3}} = \frac{26x\sqrt{3}}{\sqrt{3}\sqrt{3}} = \frac{26x\sqrt{3}}{3}\] So, our expression becomes: \[10(x^2 + \frac{26x\sqrt{3}}{3} - 105)\] Now, we need to factor the quadratic inside the parentheses. To do this, we look for two numbers that multiply to give us the product of the coefficient of \(x^2\) (which is \(1\)) and the constant term (\(-105\)), and add up to the coefficient of \(x\) (which is \(\frac{26\sqrt{3}}{3}\)). The product of the coefficient of \(x^2\) and the constant term is \((-105) \times 1 = -105\). We need to find two numbers that multiply to \(-105\) and add up to \(\frac{26\sqrt{3}}{3}\). These numbers are \(\frac{35\sqrt{3}}{3}\) and \(-\frac{30\sqrt{3}}{3}\) because: \[\frac{35\sqrt{3}}{3} \times -\frac{30\sqrt{3}}{3} = -105\] and \[\frac{35\sqrt{3}}{3} + -\frac{30\sqrt{3}}{3} = \frac{5\sqrt{3}}{3} \times 6 = \frac{30\sqrt{3}}{3} = \frac{26\sqrt{3}}{3}\] Thus, we can rewrite the quadratic as: \[x^2 + \frac{35\sqrt{3}}{3}x - \frac{30\sqrt{3}}{3}x - 105\] This can be factored by grouping: \[x(x + \frac{35\sqrt{3}}{3}) - 35(x + \frac{35\sqrt{3}}{3})\] \[= (x - 35)(x + \frac{35\sqrt{3}}{3})\] Therefore, the fully factored form of the original expression is: \[10(x - 35)(x + \frac{35\sqrt{3}}{3})\] Putting the final answer in a \(\boxed{}\): \[\boxed{10(x - 35)(x + \frac{35\sqrt{3}}{3})}\] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| baichuan4 | To factor the quadratic equation $10x^2 + \frac{260x}{\sqrt{3}} - 1050$, we first need to make sure that the coefficients are integers. We can do this by multiplying the entire equation by $\sqrt{3}$ to remove the radical from the coefficient of the $x$ term. Multiplying through by $\sqrt{3}$ gives us: $$10x^2\sqrt{3} + \frac{260x}{\sqrt{3}}\sqrt{3} - 1050\sqrt{3}$$ Simplifying, we get: $$10\sqrt{3}x^2 + 260x - 1050\sqrt{3}$$ Now, we can try to factor this quadratic using the standard form $(ax + b)(cx + d)$. To do this, we need to find two numbers whose product is $ac$ and whose sum is $b$. In this case, $a = 10\sqrt{3}$, $b = 260$, and $c = 1$. So we need to find two numbers whose product is $10\sqrt{3}$ and whose sum is $260$. We can factor $10\sqrt{3}$ as $(5\sqrt{3})(2)$. Now, we need to find two numbers that multiply to $10\sqrt{3}$ and add to $260$. By trial and error or by using the quadratic formula, we can find that these numbers are $5\sqrt{3}$ and $52$. Therefore, we can write the quadratic as: $$(5\sqrt{3}x + 52)(2x - 21)$$ This is because $(5\sqrt{3})(2) = 10\sqrt{3}$ and $(5\sqrt{3})(-21) + (52)(2) = -105\sqrt{3} + 104 = 260 - 1050/\sqrt{3}$. So, the factored form of the quadratic is: $$\boxed{(5\sqrt{3}x + 52)(2x - 21)}$$ |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| doubao | \[ \begin{align*} &10x^2+\frac{260x}{\sqrt{3}}-1050\\ =&10(x^2+\frac{26x}{\sqrt{3}}-105)\\ =&10(x^2+\frac{26\sqrt{3}x}{3}-105)\\ =&10(x^2+\frac{26\sqrt{3}x}{3}+\frac{169}{3}-\frac{169}{3}-105)\\ =&10[(x+\frac{13\sqrt{3}}{3})^2-(\frac{\sqrt{409}}{3})^2]\\ =&10(x+\frac{13\sqrt{3}}{3}+\frac{\sqrt{409}}{3})(x+\frac{13\sqrt{3}}{3}-\frac{\sqrt{409}}{3}) \end{align*} \] \[ \boxed{10(x+\frac{13\sqrt{3}}{3}+\frac{\sqrt{409}}{3})(x+\frac{13\sqrt{3}}{3}-\frac{\sqrt{409}}{3})} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| gpto1 | To factor the quadratic expression \(10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050\), we'll aim to break it down into its component factors. Here's a step-by-step approach: **Step 1: Simplify the expression by eliminating radicals in denominators.** First, to eliminate denominators, multiply both sides of the equation by \(\sqrt{3}\): \[ \sqrt{3} \times \left(10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050\right) = 0. \] Simplify: \[ 10 \sqrt{3} x^2 + 260 x - 1050 \sqrt{3} = 0. \] **Step 2: Factor the simplified quadratic expression.** We can rewrite the quadratic as: \[ 10 \sqrt{3} x^2 + 260 x - 1050 \sqrt{3} = 10 \left( \sqrt{3} x^2 + 26 x - 105 \sqrt{3} \right). \] Next, factor the quadratic inside the parentheses. To do this, consider factoring it as \((\sqrt{3} x + a)(x + b)\). Expanding the product, we get: \[ \sqrt{3} x^2 + (\sqrt{3} b + a)x + a b = \sqrt{3} x^2 + 26 x - 105 \sqrt{3}. \] Set up the system of equations: \[ \begin{cases} \sqrt{3} b + a = 26,\\ a b = -105 \sqrt{3}. \end{cases} \] Let \(a = 26 - \sqrt{3} b\). Substitute back into \(a b\): \[ (26 - \sqrt{3} b) b = -105 \sqrt{3} \implies 26 b - \sqrt{3} b^2 = -105 \sqrt{3}. \] Multiply both sides by \(-1\) to make calculations easier: \[ \sqrt{3} b^2 - 26 b + 105 \sqrt{3} = 0. \] Let \(b = k \sqrt{3}\). Substituting and simplifying, we get the quadratic equation: \[ 3 k^2 - 26 k - 105 = 0. \] Solving this quadratic equation, we find \(k = \dfrac{35}{3}\) or \(k = -3\). **Step 3: Determine the factors.** Corresponding to \(k = \dfrac{35}{3}\): \[ b = \dfrac{35}{3} \sqrt{3}, \quad a = 26 - 3 k = -9. \] Corresponding to \(k = -3\): \[ b = -3 \sqrt{3}, \quad a = 26 - 3 (-3) = 35. \] Using the solution that results in integer coefficients in \(a\) and \(b\), we select \(k = -3\). Therefore, the factors are: \[ \sqrt{3} x^2 + 26 x - 105 \sqrt{3} = (\sqrt{3} x + 35)(x - 3 \sqrt{3}). \] **Step 4: Write the final factored form.** Including the factor of \(10\) we factored out earlier, the original quadratic expression factors completely as: \[ 10 x^2 + \dfrac{260 x}{\sqrt{3}} - 1050 = 10 (\sqrt{3} x + 35)(x - 3 \sqrt{3}). \] Therefore, the factored form is: \(\boxed{10\,(\sqrt{3}x+35)\left(x-3\sqrt{3}\right)}\) |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| kimi | To factor the quadratic \(10x^2 + \frac{260x}{\sqrt{3}} - 1050\), we will follow these steps: 1. **Simplify the coefficients**: First, let's simplify the coefficient of \(x\) by rationalizing the denominator. \[ \frac{260}{\sqrt{3}} = \frac{260 \sqrt{3}}{3} \] So the quadratic becomes: \[ 10x^2 + \frac{260\sqrt{3}}{3}x - 1050 \] 2. **Find the product and sum**: To factor the quadratic, we need to find two numbers that multiply to the product of the coefficient of \(x^2\) and the constant term, and add up to the coefficient of \(x\). The product is: \[ 10 \times (-1050) = -10500 \] We need two numbers that multiply to \(-10500\) and add up to \(\frac{260\sqrt{3}}{3}\). 3. **Rationalize the quadratic**: To make it easier to find these numbers, we can multiply the entire quadratic by 3 to clear the fraction: \[ 3 \left(10x^2 + \frac{260\sqrt{3}}{3}x - 1050\right) = 30x^2 + 260\sqrt{3}x - 3150 \] Now, we need to find two numbers that multiply to \(30 \times (-3150) = -94500\) and add up to \(260\sqrt{3}\). 4. **Find the numbers**: We can use the quadratic formula to find the roots of the quadratic equation \(30x^2 + 260\sqrt{3}x - 3150 = 0\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 30\), \(b = 260\sqrt{3}\), and \(c = -3150\). Plugging in these values, we get: \[ x = \frac{-260\sqrt{3} \pm \sqrt{(260\sqrt{3})^2 - 4 \cdot 30 \cdot (-3150)}}{2 \cdot 30} \] Simplifying inside the square root: \[ x = \frac{-260\sqrt{3} \pm \sqrt{202800 + 378000}}{60} = \frac{-260\sqrt{3} \pm \sqrt{580800}}{60} = \frac{-260\sqrt{3} \pm 762}{60} \] This gives us two solutions: \[ x = \frac{-260\sqrt{3} + 762}{60} = \frac{-130\sqrt{3} + 381}{30} \] and \[ x = \frac{-260\sqrt{3} - 762}{60} = \frac{-130\sqrt{3} - 381}{30} \] 5. **Write the factored form**: The quadratic can be factored as: \[ 30(x - \frac{-130\sqrt{3} + 381}{30})(x - \frac{-130\sqrt{3} - 381}{30}) \] Simplifying the factors: \[ 30 \left( x + \frac{130\sqrt{3} - 381}{30} \right) \left( x + \frac{130\sqrt{3} + 381}{30} \right) \] This can be written as: \[ \left( \sqrt{3}x + 13 \right) \left( 10x - 30\sqrt{3} \right) \] 6. **Final answer**: The factored form of the quadratic is: \[ \boxed{(\sqrt{3}x + 13)(10x - 30\sqrt{3})} \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| minimax | To factor the quadratic $10x^2 + \frac{260x}{\sqrt{3}} - 1050$, we first need to make the coefficients more manageable by eliminating the fraction. We can do this by multiplying the entire equation by $\sqrt{3}$ to get rid of the denominator: $\sqrt{3} \cdot 10x^2 + \sqrt{3} \cdot \frac{260x}{\sqrt{3}} - \sqrt{3} \cdot 1050 = 0$ This simplifies to: $10\sqrt{3}x^2 + 260x - 1050\sqrt{3} = 0$ Now, we can factor out the greatest common factor (GCF) from the quadratic, which is 2 in this case: $2(5\sqrt{3}x^2 + 130x - 525\sqrt{3}) = 0$ Next, we look for two numbers that multiply to $5\sqrt{3} \times -525\sqrt{3} = -2625 \times 3$ and add up to |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| wenxinyiyan | To factor the given quadratic, we first need to find two numbers that multiply to the constant term (-1050) and add to the coefficient of the linear term ($\frac{260}{\sqrt{3}}$). Let's first simplify the coefficient of the linear term: $\frac{260}{\sqrt{3}} = \frac{260\sqrt{3}}{3}$ Now, we need to find two numbers that multiply to -1050 and add to $\frac{260\sqrt{3}}{3}$. After some trial and error, we find that the two numbers are $70\sqrt{3}$ and $-\frac{30}{\sqrt{3}}$. Now, we can factor the quadratic as follows: $10x^2 + \frac{260x}{\sqrt{3}} - 1050 = 10(x^2 + \frac{26x}{\sqrt{3}} - 105)$ $= 10(x + 7\sqrt{3})(x - \frac{30}{\sqrt{3}})$ However, we can further simplify the second factor: $x - \frac{30}{\sqrt{3}} = x - 10\sqrt{3}$ So, the final factored form of the quadratic is: $\boxed{10(x + 7\sqrt{3})(x - 10\sqrt{3})}$ |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| zhipu | First, we factor out the common factor of 10 from each term: \[ 10 x^2+\frac{260 x}{\sqrt{3}}-1050 = 10(x^2+\frac{26 x}{\sqrt{3}}-105). \] Next, we focus on factoring the quadratic inside the parentheses. We look for two numbers that multiply to $-105$ and add to $\frac{26}{\sqrt{3}}$. To rationalize the coefficients, multiply the entire quadratic by $\sqrt{3}$: \[ \sqrt{3} \cdot (x^2+\frac{26 x}{\sqrt{3}}-105) = \sqrt{3}x^2 + 26x - 105\sqrt{3}. \] Now, factor the quadratic $\sqrt{3}x^2 + 26x - 105\sqrt{3}$. We need two numbers that multiply to $-105\sqrt{3} \cdot \sqrt{3} = -315$ and add to 26. These numbers are 35 and -9: \[ \sqrt{3}x^2 + 26x - 105\sqrt{3} = (\sqrt{3}x + 35)(x - 3\sqrt{3}). \] Divide back by $\sqrt{3}$ to return to the original coefficients: \[ x^2+\frac{26 x}{\sqrt{3}}-105 = \left(x + \frac{35}{\sqrt{3}}\right)\left(x - 3\sqrt{3}\right). \] Now, include the factored out 10: \[ 10(x^2+\frac{26 x}{\sqrt{3}}-105) = 10\left(x + \frac{35}{\sqrt{3}}\right)\left(x - 3\sqrt{3}\right). \] To simplify further, rationalize the denominator in the first factor: \[ x + \frac{35}{\sqrt{3}} = x + \frac{35\sqrt{3}}{3}. \] Thus, the final factorization is: \[ \boxed{10\left(x + \frac{35\sqrt{3}}{3}\right)(x - 3\sqrt{3})}. \] |
10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-57f89e98-c690-92e6-8140-f7eb3a5a4987', 'request_id': '57f89e98-c690-92e6-8140-f7eb3a5a4987'} | 10 \left(-x-\frac{35}{\sqrt{3}}\right) \left(3 \sqrt{3}-x\right) |