| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B. \sqrt{2} |
C |
| deepseekr1 | To solve the quadratic equation \(2x^2 + 5x + 12 = 19 - 7x\), first rearrange all terms to one side: \[ 2x^2 + 5x + 12 - 19 + 7x = 0 \implies 2x^2 + 12x - 7 = 0. \] Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 12\), and \(c = -7\): 1. Calculate the discriminant: \[ \Delta = 12^2 - 4(2)(-7) = 144 + 56 = 200. \] \(\sqrt{200} = 10\sqrt{2}\). 2. Substitute into the formula: \[ x = \frac{-12 \pm 10\sqrt{2}}{4} = \frac{-6 \pm 5\sqrt{2}}{2}. \] The solutions are \(\frac{-6 + 5\sqrt{2}}{2}\) and \(\frac{-6 - 5\sqrt{2}}{2}\). The positive difference between the roots is: \[ \left|\frac{-6 + 5\sqrt{2}}{2} - \frac{-6 - 5\sqrt{2}}{2}\right| = \frac{10\sqrt{2}}{2} = 5\sqrt{2}. \] **Answer:** C. \(5\sqrt{2}\) |
C |
| gpto1 | C | C |
| GPT-4o | To find the positive difference between the solutions of the quadratic equation \(2x^2 + 5x + 12 = 19 - 7x\), we start by rewriting the equation in standard form. First, simplify the equation: \[2x^2 + 5x + 12 = 19 - 7x\] Move all terms to one side to set the equation to zero: \[2x^2 + 5x + 12 - 19 + 7x = 0\] Combine like terms: \[2x^2 + 12x - 7 = 0\] Now, we have the quadratic equation in standard form: \[2x^2 + 12x - 7 = 0\] To find the positive difference between the solutions, we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] For the given quadratic equation \(2x^2 + 12x - 7 = 0\), the coefficients are: \(a = 2\), \(b = 12\), and \(c = -7\). First, find the discriminant: \[\Delta = b^2 - 4ac\] \[\Delta = 12^2 - 4(2)(-7)\] \[\Delta = 144 + 56\] \[\Delta = 200\] Next, use the quadratic formula to find the roots: \[x = \frac{-12 \pm \sqrt{200}}{2 \cdot 2}\] \[x = \frac{-12 \pm \sqrt{200}}{4}\] \[x = \frac{-12 \pm 10\sqrt{2}}{4}\] \[x = \frac{-12}{4} \pm \frac{10\sqrt{2}}{4}\] \[x = -3 \pm \frac{5\sqrt{2}}{2}\] The solutions are: \[x_1 = -3 + \frac{5\sqrt{2}}{2}\] \[x_2 = -3 - \frac{5\sqrt{2}}{2}\] To find the positive difference between these two solutions: \[|x_1 - x_2| = \left| \left(-3 + \frac{5\sqrt{2}}{2}\right) - \left(-3 - \frac{5\sqrt{2}}{2}\right) \right|\] \[|x_1 - x_2| = \left| -3 + \frac{5\sqrt{2}}{2} + 3 + \frac{5\sqrt{2}}{2} \right|\] \[|x_1 - x_2| = \left| \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} \right|\] \[|x_1 - x_2| = \left| \frac{10\sqrt{2}}{2} \right|\] \[|x_1 - x_2| = 5\sqrt{2}\] Thus, the positive difference between the solutions is: \[C. 5\sqrt{2}\] |
C |
| llama3.1-70B-Instruct | C |
C |
| llama3.1-405B-Instruct | C The best answer is C |
C |
| DeepSeek-Coder-V2 | To solve the quadratic equation \(2x^2 + 5x + 12 = 19 - 7x\), we first move all terms to one side to set the equation to zero: \[ 2x^2 + 5x + 12 - 19 + 7x = 0 \] Simplify the equation: \[ 2x^2 + 12x - 7 = 0 \] Next, we use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 12\), and \(c = -7\). Calculate the discriminant \(\Delta = b^2 - 4ac\): \[ \Delta = 12^2 - 4 \cdot 2 \cdot (-7) = 144 + 56 = 200 \] Now, apply the quadratic formula: \[ x = \frac{-12 \pm \sqrt{200}}{4} = \frac{-12 \pm 10\sqrt{2}}{4} = \frac{-12}{4} \pm \frac{10\sqrt{2}}{4} = -3 \pm \frac{5\sqrt{2}}{2} \] The solutions are: \[ x_1 = -3 + \frac{5\sqrt{2}}{2} \quad \text{and} \quad x_2 = -3 - \frac{5\sqrt{2}}{2} \] To find the positive difference between these solutions: \[ \left| \left(-3 + \frac{5\sqrt{2}}{2}\right) - \left(-3 - \frac{5\sqrt{2}}{2}\right) \right| = \left| \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} \right| = \left| \frac{10\sqrt{2}}{2} \right| = 5\sqrt{2} \] Thus, the positive difference between the solutions is: \[ \boxed{C} \] |
C |
| Qwen2-72B-Instruct | First, we simplify the quadratic equation by moving all terms to one side: \[2x^2 + 5x + 12 = 19 - 7x\] \[2x^2 + 12x - 7 = 0\] We can use the quadratic formula to find the solutions for x: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\] Here, \(a = 2\), \(b = 12\), and \(c = -7\). Plugging these values into the formula gives: \[x = \frac{-12 \pm \sqrt{12^2 - 4*2*(-7)}}{2*2}\] \[x = \frac{-12 \pm \sqrt{144 + 56}}{4}\] \[x = \frac{-12 \pm \sqrt{200}}{4}\] \[x = \frac{-12 \pm 10\sqrt{2}}{4}\] \[x = \frac{-12}{4} \pm \frac{10\sqrt{2}}{4}\] \[x = -3 \pm \frac{5\sqrt{2}}{2}\] So, the two solutions are: \[x_1 = -3 + \frac{5\sqrt{2}}{2}\] \[x_2 = -3 - \frac{5\sqrt{2}}{2}\] The positive difference between these solutions is: \[x_1 - x_2 = \left(-3 + \frac{5\sqrt{2}}{2}\right) - \left(-3 - \frac{5\sqrt{2}}{2}\right)\] \[= \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}\] \[= 5\sqrt{2}\] Therefore, the answer is: C. \(5\sqrt{2}\) |
C |
| baichuan4 | C | C |
| doubao | First, we need to transform the equation into the standard form of a quadratic equation: $2x^2 + 5x + 7x + 12 - 19 = 0$, which simplifies to $2x^2 + 12x - 7 = 0$. Using the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 2$, $b = 12$, and $c = -7$. The discriminant is $12^2 - 4 \times 2 \times (-7) = 144 + 56 = 200$. The two solutions are $x_1 = \frac{-12 + \sqrt{200}}{4} = \frac{-12 + 10\sqrt{2}}{4} = \frac{-3 + 5\sqrt{2}}{2}$ and $x_2 = \frac{-12 - \sqrt{200}}{4} = \frac{-12 - 10\sqrt{2}}{4} = \frac{-3 - 5\sqrt{2}}{2}$. The positive difference between the solutions is $(\frac{-3 + 5\sqrt{2}}{2}) - (\frac{-3 - 5\sqrt{2}}{2}) = 5\sqrt{2}$. So the answer is C. |
C |
| minimax | First, let's solve the quadratic equation by rearranging it into standard form: \[2x^2 + 5x + 12 = 19 - 7x\] Adding \(7x\) to both sides and subtracting \(19\) from both sides gives: \[2x^2 + 12x - 7 = 0\] Now we can use the quadratic formula to find the solutions for \(x\), where \(a = 2\), \(b = 12\), and \(c = -7\): \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Plugging in the values, we get: \[x = \frac{-12 \pm \sqrt{12^2 - 4(2)(-7)}}{2(2)}\] \[x = \frac{-12 \pm \sqrt{144 + 56}}{4}\] \[x = \frac{-12 \pm \sqrt{200}}{4}\] \[x = \frac{-12 \pm 10\sqrt{2}}{4}\] Now we can simplify the solutions: \[x_1 = \frac{-12 + 10\sqrt{2}}{4} = -3 + \frac{5\sqrt{2}}{2}\] \[x_2 = \frac{-12 - 10\sqrt{2}}{4} = -3 - \frac{5\sqrt{2}}{2}\] The positive difference between these two solutions is: \[x_1 - x_2 = \left(-3 + \frac{5\sqrt{2}}{2}\right) - \left(-3 - \frac{5\sqrt{2}}{2}\right)\] \[x_1 - x_2 = -3 + \frac{5\sqrt{2}}{2} + 3 + \frac{5\sqrt{2}}{2}\] \[x_1 - x_2 = \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}\] \[x_1 - x_2 = 5\sqrt{2}\] So the positive difference between the two solutions is \(5\sqrt{2}\), which corresponds to answer: C. \(5\sqrt{2}\) |
C |
| kimi | First, let's simplify the equation by moving all terms to one side: \[2x^2 + 5x + 12 = 19 - 7x\] \[2x^2 + 5x + 7x + 12 - 19 = 0\] \[2x^2 + 12x - 7 = 0\] Now, we can divide the entire equation by 2 to simplify it further: \[x^2 + 6x - \frac{7}{2} = 0\] Next, we need to find the solutions to this quadratic equation. We can use the quadratic formula, which is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] For our equation, \(a = 1\), \(b = 6\), and \(c = -\frac{7}{2}\). Plugging these values into the quadratic formula gives us: \[x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot \left(-\frac{7}{2}\right)}}{2 \cdot 1}\] \[x = \frac{-6 \pm \sqrt{36 + 14}}{2}\] \[x = \frac{-6 \pm \sqrt{50}}{2}\] \[x = \frac{-6 \pm 5\sqrt{2}}{2}\] \[x = -3 \pm \frac{5\sqrt{2}}{2}\] The two solutions are \(x_1 = -3 + \frac{5\sqrt{2}}{2}\) and \(x_2 = -3 - \frac{5\sqrt{2}}{2}\). The positive difference between these solutions is: \[x_1 - x_2 = \left(-3 + \frac{5\sqrt{2}}{2}\right) - \left(-3 - \frac{5\sqrt{2}}{2}\right)\] \[x_1 - x_2 = \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2}\] \[x_1 - x_2 = 5\sqrt{2}\] So the answer is C. \(5\sqrt{2}\). |
C |
| wenxinyiyan | C | C |
| zhipu | First, we need to bring the equation to standard quadratic form. Starting with the given equation: \[ 2x^2 + 5x + 12 = 19 - 7x \] Add \( 7x \) to both sides and subtract \( 19 \) from both sides to get: \[ 2x^2 + 12x - 7 = 0 \] Now, we solve the quadratic equation \( 2x^2 + 12x - 7 = 0 \) using the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 2 \), \( b = 12 \), and \( c = -7 \). Plugging in these values: \[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 2 \cdot (-7)}}{2 \cdot 2} \] Calculate the discriminant: \[ 12^2 - 4 \cdot 2 \cdot (-7) = 144 + 56 = 200 \] So, the quadratic formula becomes: \[ x = \frac{-12 \pm \sqrt{200}}{4} \] Simplify \( \sqrt{200} \) to \( 10\sqrt{2} \): \[ x = \frac{-12 \pm 10\sqrt{2}}{4} \] Divide each term by 4: \[ x = \frac{-12}{4} \pm \frac{10\sqrt{2}}{4} \] \[ x = -3 \pm \frac{5\sqrt{2}}{2} \] The two solutions are: \[ x_1 = -3 + \frac{5\sqrt{2}}{2} \] \[ x_2 = -3 - \frac{5\sqrt{2}}{2} \] To find the positive difference between these solutions: \[ \left| x_1 - x_2 \right| = \left| \left( -3 + \frac{5\sqrt{2}}{2} \right) - \left( -3 - \frac{5\sqrt{2}}{2} \right) \right| \] \[ = \left| \frac{5\sqrt{2}}{2} + \frac{5\sqrt{2}}{2} \right| \] \[ = \left| 5\sqrt{2} \right| \] \[ = 5\sqrt{2} \] Therefore, the positive difference between the solutions is: \[ \boxed{5\sqrt{2}} \] So the correct answer is: C. \( 5\sqrt{2} \) |
C |
| gpto1 | C | C |
| deepseekv3 | server error: Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-4f22f9db-9980-9cc5-87d5-6ca3a7b98518', 'request_id': '4f22f9db-9980-9cc5-87d5-6ca3a7b98518'} | C |