| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To factor the quadratic expression \(9x^2 + 27\sqrt{7}x - 3402\), we will follow these steps: 1. **Identify the coefficients**: The given quadratic is in the form \(ax^2 + bx + c\), where: - \(a = 9\) - \(b = 27\sqrt{7}\) - \(c = -3402\) 2. **Factor out the common factor**: Notice that all terms have a common factor of 9. We can factor out 9 from each term: \[ 9x^2 + 27\sqrt{7}x - 3402 = 9(x^2 + 3\sqrt{7}x - 378) \] 3. **Factor the remaining quadratic**: Now, we need to factor the quadratic expression inside the parentheses: \[ x^2 + 3\sqrt{7}x - 378 \] 4. **Find the roots of the quadratic equation**: To factor \(x^2 + 3\sqrt{7}x - 378\), we need to find two numbers whose product is \(-378\) and whose sum is \(3\sqrt{7}\). 5. **Use the quadratic formula**: The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For our quadratic \(x^2 + 3\sqrt{7}x - 378\), \(a = 1\), \(b = 3\sqrt{7}\), and \(c = -378\). Plugging these values into the quadratic formula: \[ x = \frac{-3\sqrt{7} \pm \sqrt{(3\sqrt{7})^2 - 4 \cdot 1 \cdot (-378)}}{2 \cdot 1} \] Simplify under the square root: \[ x = \frac{-3\sqrt{7} \pm \sqrt{63 + 1512}}{2} \] \[ x = \frac{-3\sqrt{7} \pm \sqrt{1575}}{2} \] \[ x = \frac{-3\sqrt{7} \pm 15\sqrt{7}}{2} \] This gives us two solutions: \[ x = \frac{-3\sqrt{7} + 15\sqrt{7}}{2} = \frac{12\sqrt{7}}{2} = 6\sqrt{7} \] \[ x = \frac{-3\sqrt{7} - 15\sqrt{7}}{2} = \frac{-18\sqrt{7}}{2} = -9\sqrt{7} \] 6. **Form the factored expression**: The roots \(6\sqrt{7}\) and \(-9\sqrt{7}\) give us the factors: \[ (x - 6\sqrt{7})(x + 9\sqrt{7}) \] 7. **Combine with the common factor**: Recall that we factored out 9 earlier, so the complete factored form is: \[ 9(x - 6\sqrt{7})(x + 9\sqrt{7}) \] Thus, the factored form of the quadratic expression is: \[ \boxed{9(x - 6\sqrt{7})(x + 9\sqrt{7})} \] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| deepseek-r1 | To factor the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), we start by factoring out the greatest common factor, which is 9: \[ 9x^2 + 27\sqrt{7}x - 3402 = 9(x^2 + 3\sqrt{7}x - 378) \] Next, we need to factor the quadratic expression inside the parentheses: \(x^2 + 3\sqrt{7}x - 378\). We look for two numbers that multiply to \(-378\) and add up to \(3\sqrt{7}\). We assume these numbers are of the form \(a\sqrt{7}\) and \(b\sqrt{7}\) where \(a\) and \(b\) are integers. This gives us the equations: 1. \(a + b = 3\) 2. \(7ab = -378\) Solving the second equation, we find \(ab = -54\). We need integers \(a\) and \(b\) such that their product is \(-54\) and their sum is 3. The pair \(-6\) and \(9\) satisfies these conditions: - \(a = -6\) and \(b = 9\) (or vice versa) Thus, the numbers are \(-6\sqrt{7}\) and \(9\sqrt{7}\). The quadratic expression factors as: \[ (x - 6\sqrt{7})(x + 9\sqrt{7}) \] Multiplying this result by the previously factored out 9, we get the final factorization: \[ 9(x - 6\sqrt{7})(x + 9\sqrt{7}) \] The factored form of the quadratic is \(\boxed{9(x - 6\sqrt{7})(x + 9\sqrt{7})}\). |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| gpto1 | To factor the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), we begin by factoring out the greatest common factor, which is 9: \[9x^2 + 27\sqrt{7}x - 3402 = 9(x^2 + 3\sqrt{7}x - 378).\] Next, we focus on factoring the quadratic inside the parentheses: \(x^2 + 3\sqrt{7}x - 378\). To factor this quadratic, we can either look for two numbers that multiply to -378 and add to \(3\sqrt{7}\) (which is challenging due to the square roots) or directly use the quadratic formula to find the roots: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where \(a = 1\), \(b = 3\sqrt{7}\), and \(c = -378\). Calculating the discriminant: \[D = b^2 - 4ac = (3\sqrt{7})^2 - 4(1)(-378) = 63 + 1512 = 1575.\] Simplify \(\sqrt{1575}\): \[\sqrt{1575} = \sqrt{25 \times 9 \times 7} = 15\sqrt{7}.\] Compute the roots: \[x = \frac{-3\sqrt{7} \pm 15\sqrt{7}}{2}.\] So the roots are: \[ \begin{align*} x &= \frac{-3\sqrt{7} + 15\sqrt{7}}{2} = \frac{12\sqrt{7}}{2} = 6\sqrt{7}, \\ x &= \frac{-3\sqrt{7} - 15\sqrt{7}}{2} = \frac{-18\sqrt{7}}{2} = -9\sqrt{7}. \end{align*} \] Therefore, the quadratic factors to: \[x^2 + 3\sqrt{7}x - 378 = (x - 6\sqrt{7})(x + 9\sqrt{7}).\] Putting it all together: \[9x^2 + 27\sqrt{7}x - 3402 = 9(x - 6\sqrt{7})(x + 9\sqrt{7}) = (x - 6\sqrt{7})(9x + 81\sqrt{7}).\] So, the fully factored form is: \[\boxed{(x\ -\ 6\sqrt{7})\,\left(9x\ +\ 81\sqrt{7}\right)}\] \boxed{(x\ -\ 6\sqrt{7})\,\left(9x\ +\ 81\sqrt{7}\right)} |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| gpt-4o-2024-08-06 | To factor the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), we will use the quadratic formula to find its roots. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] For the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), the coefficients are: - \(a = 9\) - \(b = 27\sqrt{7}\) - \(c = -3402\) First, we calculate the discriminant \(b^2 - 4ac\): \[ b^2 = (27\sqrt{7})^2 = 729 \times 7 = 5103 \] \[ 4ac = 4 \times 9 \times (-3402) = -122472 \] \[ b^2 - 4ac = 5103 + 122472 = 127575 \] Next, we find the square root of the discriminant: \[ \sqrt{127575} \] To simplify \(\sqrt{127575}\), we factor 127575: \[ 127575 = 3^2 \times 5^2 \times 7 \times 13^2 \] Thus, \[ \sqrt{127575} = \sqrt{3^2 \times 5^2 \times 7 \times 13^2} = 3 \times 5 \times 13 \times \sqrt{7} = 195\sqrt{7} \] Now, substitute back into the quadratic formula: \[ x = \frac{-27\sqrt{7} \pm 195\sqrt{7}}{18} \] This gives us two solutions: 1. \(x_1 = \frac{-27\sqrt{7} + 195\sqrt{7}}{18} = \frac{168\sqrt{7}}{18} = \frac{28\sqrt{7}}{3}\) 2. \(x_2 = \frac{-27\sqrt{7} - 195\sqrt{7}}{18} = \frac{-222\sqrt{7}}{18} = -\frac{37\sqrt{7}}{3}\) Thus, the roots of the quadratic are \(x = \frac{28\sqrt{7}}{3}\) and \(x = -\frac{37\sqrt{7}}{3}\). The factored form of the quadratic is: \[ 9(x - \frac{28\sqrt{7}}{3})(x + \frac{37\sqrt{7}}{3}) \] Simplifying, we multiply each term by 3 to clear the fractions: \[ = 9 \left( \frac{3x - 28\sqrt{7}}{3} \right) \left( \frac{3x + 37\sqrt{7}}{3} \right) \] \[ = (3x - 28\sqrt{7})(3x + 37\sqrt{7}) \] Thus, the factored form of the quadratic is: \[ \boxed{(3x - 28\sqrt{7})(3x + 37\sqrt{7})} \] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| DeepSeek-Coder-V2-Lite-Instruct | To factor the quadratic expression \(9 x^2 + 27 \sqrt{7} x - 3402\), we will follow these steps: 1. **Identify the coefficients**: The quadratic is in the form \(ax^2 + bx + c\), where \(a = 9\), \(b = 27 \sqrt{7}\), and \(c = -3402\). 2. **Find the product \(ac\)**: \[ ac = 9 \times (-3402) = -30618 \] 3. **Find factors of \(ac\) that add up to \(b\)**: We need to find two numbers that multiply to \(-30618\) and add up to \(27 \sqrt{7}\). 4. **Identify the correct factors**: The factors of \(-30618\) that add up to \(27 \sqrt{7}\) are not straightforward to find directly. Instead, we can use the quadratic formula to find the roots and then factor the quadratic based on those roots. 5. **Use the quadratic formula**: The roots of the quadratic equation \(ax^2 + bx + c = 0\) are given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substituting \(a = 9\), \(b = 27 \sqrt{7}\), and \(c = -3402\): \[ x = \frac{-27 \sqrt{7} \pm \sqrt{(27 \sqrt{7})^2 - 4 \cdot 9 \cdot (-3402)}}{2 \cdot 9} \] \[ x = \frac{-27 \sqrt{7} \pm \sqrt{729 \cdot 7 + 122496}}{18} \] \[ x = \frac{-27 \sqrt{7} \pm \sqrt{5103 + 122496}}{18} \] \[ x = \frac{-27 \sqrt{7} \pm \sqrt{127599}}{18} \] \[ x = \frac{-27 \sqrt{7} \pm 357 \sqrt{7}}{18} \] \[ x = \frac{-27 \sqrt{7} + 357 \sqrt{7}}{18} \quad \text{or} \quad x = \frac{-27 \sqrt{7} - 357 \sqrt{7}}{18} \] \[ x = \frac{330 \sqrt{7}}{18} \quad \text{or} \quad x = \frac{-384 \sqrt{7}}{18} \] \[ x = 18.33 \sqrt{7} \quad \text{or} \quad x = -21.33 \sqrt{7} \] 6. **Factor the quadratic**: The roots are \(x = 18.33 \sqrt{7}\) and \(x = -21.33 \sqrt{7}\). Therefore, the quadratic can be factored as: \[ 9 x^2 + 27 \sqrt{7} x - 3402 = 9 \left(x - 18.33 \sqrt{7}\right) \left(x + 21.33 \sqrt{7}\right) \] However, this factorization is not in the simplest form. Let's re-evaluate the factors of \(ac\) that add up to \(b\) more carefully. 7. **Revisit the factors**: Let's try to find simpler factors. We can check if the quadratic can be factored by grouping or by recognizing a simpler factorization. 8. **Simplified factorization**: After re-evaluating, we find that the quadratic can be factored as: \[ 9 x^2 + 27 \sqrt{7} x - 3402 = 9 \left(x - 7 \sqrt{7}\right) \left(x + 18 \sqrt{7}\right) \] Thus, the final factored form is: \[ \boxed{9 \left(x - 7 \sqrt{7}\right) \left(x + 18 \sqrt{7}\right)} \] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the quadratic equation The given quadratic equation is $9x^2 + 27\sqrt{7}x - 3402$. ## Step 2: Determine the coefficients The coefficients of the quadratic equation are $a = 9$, $b = 27\sqrt{7}$, and $c = -3402$. ## Step 3: Use the quadratic formula to find the roots Since the quadratic expression does not factor easily, we can use the quadratic formula to find the roots. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. ## Step 4: Plug in the values into the quadratic formula Substituting the values of $a$, $b$, and $c$ into the quadratic formula, we get $x = \frac{-27\sqrt{7} \pm \sqrt{(27\sqrt{7})^2 - 4(9)(-3402)}}{2(9)}$. ## Step 5: Simplify the expression under the square root Simplifying the expression under the square root, we get $x = \frac{-27\sqrt{7} \pm \sqrt{729(7) + 122472}}{18}$. ## Step 6: Continue simplifying the expression under the square root Continuing to simplify, we have $x = \frac{-27\sqrt{7} \pm \sqrt{5093 + 122472}}{18}$, which simplifies to $x = \frac{-27\sqrt{7} \pm \sqrt{127565}}{18}$. ## Step 7: Further simplify the expression under the square root Further simplification yields $x = \frac{-27\sqrt{7} \pm 357}{18}$. ## Step 8: Solve for the roots Solving for the roots, we get two possible values for $x$: $x = \frac{-27\sqrt{7} + 357}{18}$ and $x = \frac{-27\sqrt{7} - 357}{18}$. ## Step 9: Simplify the roots Simplifying the roots, we find $x = \frac{357 - 27\sqrt{7}}{18}$ and $x = \frac{-357 - 27\sqrt{7}}{18}$. ## Step 10: Factor the quadratic using the roots Using the roots, we can factor the quadratic as $(3x - (19 - 3\sqrt{7}))(3x - (-19 - 3\sqrt{7}))$. ## Step 11: Simplify the factored form Simplifying the factored form, we get $(3x - 19 + 3\sqrt{7})(3x + 19 + 3\sqrt{7})$. ## Step 12: Write the factored form as a product of two binomials The factored form can be written as $(3x - 19 + 3\sqrt{7})(3x + 19 + 3\sqrt{7}) = (3x + 3\sqrt{7} - 19)(3x + 3\sqrt{7} + 19)$. ## Step 13: Factor out the greatest common factor Factoring out the greatest common factor, we get $(3(x + \sqrt{7}) - 19)(3(x + \sqrt{7}) + 19)$. ## Step 14: Write the factored form as a product of two binomials with a common factor The factored form can be written as $((3(x + \sqrt{7}))^2 - 19^2)$. ## Step 15: Simplify the expression Simplifying the expression, we get $(3(x + \sqrt{7}))^2 - 361$. ## Step 16: Factor the difference of squares Factoring the difference of squares, we get $(3(x + \sqrt{7}) - 19)(3(x + \sqrt{7}) + 19)$. ## Step 17: Write the final factored form The final factored form is $(3x + 3\sqrt{7} - 19)(3x + 3\sqrt{7} + 19)$. The final answer is: $\boxed{(3x - 19 + 3\sqrt{7})(3x + 19 + 3\sqrt{7})}$ |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $9x^2 + 27\sqrt{7}x - 3402$. ## Step 2: Determine the coefficients of the quadratic equation The coefficients of the quadratic equation are $a = 9$, $b = 27\sqrt{7}$, and $c = -3402$. ## Step 3: Use the quadratic formula to find the roots of the equation The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Plugging in the values of $a$, $b$, and $c$, we get $x = \frac{-27\sqrt{7} \pm \sqrt{(27\sqrt{7})^2 - 4(9)(-3402)}}{2(9)}$. ## Step 4: Simplify the expression under the square root $(27\sqrt{7})^2 = 729 \cdot 7 = 5093$ and $-4(9)(-3402) = 122472$. So, the expression under the square root simplifies to $\sqrt{5093 + 122472} = \sqrt{127565}$. ## Step 5: Simplify the square root $\sqrt{127565} = 357$. ## Step 6: Simplify the quadratic formula expression $x = \frac{-27\sqrt{7} \pm 357}{18}$. ## Step 7: Factor the quadratic equation using the roots The factored form of the quadratic equation is $(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots of the equation. ## Step 8: Calculate the roots of the equation $r_1 = \frac{-27\sqrt{7} + 357}{18}$ and $r_2 = \frac{-27\sqrt{7} - 357}{18}$. ## Step 9: Simplify the roots $r_1 = \frac{357 - 27\sqrt{7}}{18}$ and $r_2 = \frac{-357 - 27\sqrt{7}}{18}$. ## Step 10: Factor the quadratic equation $9x^2 + 27\sqrt{7}x - 3402 = (3x - (19 - 3\sqrt{7}))(3x - (19 + 3\sqrt{7}))$. ## Step 11: Write the final answer in the required format $\boxed{(3x - (19 - 3\sqrt{7}))(3x - (19 + 3\sqrt{7}))}$. The final answer is: $\boxed{(3x - (19 - 3\sqrt{7}))(3x - (19 + 3\sqrt{7}))}$ |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| Qwen2-72B-Instruct | To factor the given quadratic expression, we first look for two numbers that multiply to give the product of the coefficient of \(x^2\) (which is 9) and the constant term (-3402), and add up to the coefficient of \(x\) (which is \(27\sqrt{7}\)). The product of the coefficient of \(x^2\) and the constant term is: \[9 \times -3402 = -30618\] We need to find two numbers that multiply to \(-30618\) and add up to \(27\sqrt{7}\). Let's denote these two numbers as \(m\) and \(n\). We have: \[mn = -30618\] \[m + n = 27\sqrt{7}\] To solve this system, we can use the factored form of the quadratic equation, which is: \[ax^2 + bx + c = a(x - p)(x - q)\] where \(p\) and \(q\) are the roots of the equation. In our case, \(a = 9\), \(b = 27\sqrt{7}\), and \(c = -3402\). The roots \(p\) and \(q\) can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Substituting the values, we get: \[x = \frac{-27\sqrt{7} \pm \sqrt{(27\sqrt{7})^2 - 4(9)(-3402)}}{2(9)}\] \[x = \frac{-27\sqrt{7} \pm \sqrt{567 - (-122472)}}{18}\] \[x = \frac{-27\sqrt{7} \pm \sqrt{567 + 122472}}{18}\] \[x = \frac{-27\sqrt{7} \pm \sqrt{123039}}{18}\] \[x = \frac{-27\sqrt{7} \pm 351}{18}\] So, the roots are: \[x_1 = \frac{-27\sqrt{7} + 351}{18}\] \[x_2 = \frac{-27\sqrt{7} - 351}{18}\] Thus, the factored form of the quadratic is: \[9(x - x_1)(x - x_2)\] Simplifying the roots gives us: \[x_1 = \frac{351 - 27\sqrt{7}}{18} = \frac{351}{18} - \frac{27\sqrt{7}}{18} = 19.5 - 1.5\sqrt{7}\] \[x_2 = \frac{-351 - 27\sqrt{7}}{18} = -19.5 - 1.5\sqrt{7}\] Therefore, the factored form is: \[9\left(x - (19.5 - 1.5\sqrt{7})\right)\left(x - (-19.5 - 1.5\sqrt{7})\right)\] This simplifies to: \[9(x - 19.5 + 1.5\sqrt{7})(x + 19.5 + 1.5\sqrt{7})\] Putting it in the requested format: \[\boxed{9(x - 19.5 + 1.5\sqrt{7})(x + 19.5 + 1.5\sqrt{7})}\] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| baichuan4 | To factor the quadratic equation $9x^2 + 27\sqrt{7}x - 3402$, we can use the following steps: 1. **Divide by the coefficient of $x^2$**: First, divide the entire equation by the coefficient of $x^2$, which is 9, to simplify the equation. This gives us: $$ x^2 + 3\sqrt{7}x - 378 $$ 2. **Find two numbers that multiply to the constant term and add to the coefficient of $x$**: We need to find two numbers $a$ and $b$ such that $ab = -378$ and $a + b = 3\sqrt{7}$. 3. **Guess and check**: We can guess and check possible factors of -378 until we find a pair that adds up to $3\sqrt{7}$. This may take some time, but it's a straightforward process. 4. **Identify the factors**: After some trial and error, we find that $a = 13\sqrt{7}$ and $b = -28$ work because: $$ (13\sqrt{7})(-28) = -378 $$ $$ 13\sqrt{7} - 28 = 3\sqrt{7} $$ 5. **Write the factored form**: Now that we have found $a$ and $b$, we can write the factored form of the equation $$ $$ (x + 13\sqrt{7})(x - 28) $$ 6. **Multiply by the original coefficient of $x^2$**: Finally, we need to remember to multiply the factored form by the original coefficient of $x^2$, which was 9. This gives us the final factored form: $$ 9(x + 13\sqrt{7})(x - 28) $$ So, the factored form of the quadratic equation $9x^2 + 27\sqrt{7}x - 3402$ is: $$ \boxed{9(x + 13\sqrt{7})(x - 28)} $$ |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| doubao | For the quadratic $9x^2+27\sqrt{7}x-3402$, we need to find two numbers that multiply to $9\times(-3402)=-30618$ and add up to $27\sqrt{7}$. We can factor out 9 first: $9(x^2+3\sqrt{7}x-378)$. Now we need to factor the quadratic inside the parentheses. We can't factor it easily by inspection, so we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$. Here $a=1$, $b=3\sqrt{7}$, and $c=-378$. $\Delta = (3\sqrt{7})^2 - 4\times1\times(-378) = 63 + 1512 = 1575$. $\sqrt{\Delta} = \sqrt{1575} = 15\sqrt{7}$. $x = \frac{-3\sqrt{7} \pm 15\sqrt{7}}{2}$. $x_1 = \frac{12\sqrt{7}}{2} = 6\sqrt{7}$, $x_2 = \frac{-18\sqrt{7}}{2} = -9\sqrt{7}$. So the factored form is $9(x-9\sqrt{7})(x+38\sqrt{7})$. $\boxed{9(x-9\sqrt{7})(x+38\sqrt{7})}$. |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| gpto1 | To factor the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), we begin by factoring out the greatest common factor, which is 9: \[9x^2 + 27\sqrt{7}x - 3402 = 9(x^2 + 3\sqrt{7}x - 378).\] Next, we focus on factoring the quadratic inside the parentheses: \(x^2 + 3\sqrt{7}x - 378\). To factor this quadratic, we can either look for two numbers that multiply to -378 and add to \(3\sqrt{7}\) (which is challenging due to the square roots) or directly use the quadratic formula to find the roots: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\] where \(a = 1\), \(b = 3\sqrt{7}\), and \(c = -378\). Calculating the discriminant: \[D = b^2 - 4ac = (3\sqrt{7})^2 - 4(1)(-378) = 63 + 1512 = 1575.\] Simplify \(\sqrt{1575}\): \[\sqrt{1575} = \sqrt{25 \times 9 \times 7} = 15\sqrt{7}.\] Compute the roots: \[x = \frac{-3\sqrt{7} \pm 15\sqrt{7}}{2}.\] So the roots are: \[ \begin{align*} x &= \frac{-3\sqrt{7} + 15\sqrt{7}}{2} = \frac{12\sqrt{7}}{2} = 6\sqrt{7}, \\ x &= \frac{-3\sqrt{7} - 15\sqrt{7}}{2} = \frac{-18\sqrt{7}}{2} = -9\sqrt{7}. \end{align*} \] Therefore, the quadratic factors to: \[x^2 + 3\sqrt{7}x - 378 = (x - 6\sqrt{7})(x + 9\sqrt{7}).\] Putting it all together: \[9x^2 + 27\sqrt{7}x - 3402 = 9(x - 6\sqrt{7})(x + 9\sqrt{7}) = (x - 6\sqrt{7})(9x + 81\sqrt{7}).\] So, the fully factored form is: \[\boxed{(x\ -\ 6\sqrt{7})\,\left(9x\ +\ 81\sqrt{7}\right)}\] \boxed{(x\ -\ 6\sqrt{7})\,\left(9x\ +\ 81\sqrt{7}\right)} |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| kimi | To factor the quadratic \(9x^2 + 27\sqrt{7}x - 3402\), we will follow these steps: 1. **Identify the coefficients**: The quadratic is in the form \(ax^2 + bx + c\), where \(a = 9\), \(b = 27\sqrt{7}\), and \(c = -3402\). 2. **Find the product of \(a\) and \(c\)**: \[ a \cdot c = 9 \cdot (-3402) = -30618 \] 3. **Find two numbers that multiply to \(-30618\) and add up to \(27\sqrt{7}\)**: We need to find two numbers whose product is \(-30618\) and whose sum is \(27\sqrt{7}\). Let's denote these two numbers by \(m\) and \(n\). We need: \[ m \cdot n = -30618 \quad \text{and} \quad m + n = 27\sqrt{7} \] 4. **Solve for \(m\) and \(n\)**: To find \(m\) and \(n\), we can use the quadratic formula or trial and error. However, since we are dealing with a quadratic equation, we can use the fact that the roots of the quadratic equation \(ax^2 + bx + c = 0\) are given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the values: \[ x = \frac{-27\sqrt{7} \pm \sqrt{(27\sqrt{7})^2 - 4 \cdot 9 \cdot (-3402)}}{2 \cdot 9} \] 5. **Calculate the discriminant**: \[ b^2 - 4ac = (27\sqrt{7})^2 - 4 \cdot 9 \cdot (-3402) \] \[ = 729 \cdot 7 + 4 \cdot 9 \cdot 3402 \] \[ = 5103 + 129672 \] \[ = 134775 \] 6. **Find the square root of the discriminant**: \[ \sqrt{134775} = 367 \] 7. **Calculate the roots**: \[ x = \frac{-27\sqrt{7} \pm 367}{18} \] This gives us two roots: \[ x_1 = \frac{-27\sqrt{7} + 367}{18} \] \[ x_2 = \frac{-27\sqrt{7} - 367}{18} \] 8. **Express the quadratic in factored form**: The quadratic can be factored as: \[ 9(x - x_1)(x - x_2) \] Substituting the roots: \[ 9 \left( x - \frac{-27\sqrt{7} + 367}{18} \right) \left( x - \frac{-27\sqrt{7} - 367}{18} \right) \] 9. **Simplify the factors**: \[ 9 \left( x + \frac{27\sqrt{7} - 367}{18} \right) \left( x + \frac{27\sqrt{7} + 367}{18} \right) \] \[ = \left( 3x + \frac{27\sqrt{7} - 367}{6} \right) \left( 3x + \frac{27\sqrt{7} + 367}{6} \right) \] 10. **Final answer**: \[ \boxed{(3x + 9\sqrt{7} - 122)(3x - 9\sqrt{7} + 122)} \] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| minimax | To factor the quadratic $9x^2 + 27\sqrt{7}x - 3402$, we need to find two numbers that multiply to $9 \times -3402$ and add up to $27\sqrt{7}$. First, let's calculate the product of the coefficient of $x^2$ and the constant term: $9 \times -3402 = -30618$ Now we need to find two numbers that multiply to $-30618$ and add up to $27\sqrt{7}$. Since $27\sqrt{7}$ is not a rational number, we can't use the standard factoring techniques that rely on finding integer pairs. However, we can still try to factor by grouping or looking for a pattern. Let's look for two numbers that multiply to $-30618$ and have a sum that is close to $27\sqrt{7}$. We can start by factoring $-30618$: $-30618 = -2 \times 15309$ Now, we need to find two factors of $1530 |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| wenxinyiyan | To factor the given quadratic, we first look for two numbers whose product is $9 \times -3402 = -30618$ and whose sum is $27\sqrt{7}$. These two numbers turn out to be $81\sqrt{7}$ and $-38\sqrt{7}$. Next, we split the middle term of the quadratic using these two numbers: $9x^2 + 81\sqrt{7}x - 38\sqrt{7}x - 3402$. Then, we factor by grouping: $9x(x + 9\sqrt{7}) - 38\sqrt{7}(x + 9\sqrt{7})$. Finally, we extract the common factor $(x + 9\sqrt{7})$: $\boxed{(9x - 38\sqrt{7})(x + 9\sqrt{7})}$. |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| zhipu | To factor the quadratic $9x^2 + 27\sqrt{7}x - 3402$, we can start by looking for two numbers that multiply to $9 \times -3402$ and add up to $27\sqrt{7}$. First, let's simplify the multiplication of the leading coefficient and the constant term: \[ 9 \times -3402 = -30618. \] Now we need to find two numbers that multiply to $-30618$ and add up to $27\sqrt{7}$. Since $27\sqrt{7}$ is not a simple integer, we should consider the possibility that the factors might involve $\sqrt{7}$. Let's assume that the quadratic can be factored as: \[ 9x^2 + 27\sqrt{7}x - 3402 = (3x + a\sqrt{7})(3x + b\sqrt{7}), \] where $a$ and $b$ are integers. Expanding the right-hand side, we get: \[ (3x + a\sqrt{7})(3x + b\sqrt{7}) = 9x^2 + (3a\sqrt{7} + 3b\sqrt{7})x + (a\sqrt{7})(b\sqrt{7}). \] This simplifies to: \[ 9x^2 + (3a + 3b)\sqrt{7}x + 7ab. \] Now we can match the coefficients: \[ 3a + 3b = 27 \quad \text{and} \quad 7ab = -3402. \] From the first equation, we can factor out the 3: \[ a + b = 9. \] From the second equation, we can solve for $ab$: \[ ab = \frac{-3402}{7} = -486. \] Now we need to find two numbers $a$ and $b$ such that $a + b = 9$ and $ab = -486$. We can look for the factors of $-486$ that add up to 9. The factors of $-486$ are $\pm1, \pm2, \pm3, \pm6, \pm9, \pm18, \pm27, \pm54, \pm81, \pm162, \pm243, \pm486$. By trial and error, we find that $-18$ and $27$ are the numbers that satisfy both conditions: \[ -18 + 27 = 9 \quad \text{and} \quad (-18)(27) = -486. \] So, $a = -18$ and $b = 27$. Therefore, the factored form of the quadratic is: \[ 9x^2 + 27\sqrt{7}x - 3402 = (3x - 18\sqrt{7})(3x + 27\sqrt{7}). \] We can simplify this further by factoring out a $3\sqrt{7}$ from each term: \[ (3x - 18\sqrt{7})(3x + 27\sqrt{7}) = 3\sqrt{7}(x - 6\sqrt{7})3\sqrt{7}(x + 9\sqrt{7}). \] This simplifies to: \[ (3\sqrt{7}x - 54\sqrt{7})(3\sqrt{7}x + 81\sqrt{7}). \] Since $3\sqrt{7}x$ is a common factor, we can write it as: \[ (3\sqrt{7}x - 54\sqrt{7})(3\sqrt{7}x + 81\sqrt{7}) = (3\sqrt{7}x - 54)(3\sqrt{7}x + 81). \] So the final factored form is: \[ \boxed{(3\sqrt{7}x - 54)(3\sqrt{7}x + 81)}. \] |
9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-ca171d7d-dc88-9070-9735-b40e8bbe72bd', 'request_id': 'ca171d7d-dc88-9070-9735-b40e8bbe72bd'} | 9 \left(-x-9 \sqrt{7}\right) \left(6 \sqrt{7}-x\right) |