| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To complete the square for the quadratic expression \(-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}\), we follow these steps: 1. **Factor out the common factor \(-\sqrt{3}\) from the quadratic and linear terms:** \[ -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} = -\sqrt{3} (x^2 - 7x + 9) \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-7\), divide it by 2, and square it: \[ \left(\frac{-7}{2}\right)^2 = \frac{49}{4} \] - Add and subtract \(\frac{49}{4}\) inside the parentheses: \[ x^2 - 7x + 9 = x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 9 \] - Rewrite the expression as a perfect square and a constant: \[ x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 9 = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 9 \] - Simplify the constant term: \[ - \frac{49}{4} + 9 = - \frac{49}{4} + \frac{36}{4} = - \frac{13}{4} \] 3. **Combine the completed square and the constant term:** \[ x^2 - 7x + 9 = \left(x - \frac{7}{2}\right)^2 - \frac{13}{4} \] 4. **Multiply the entire expression by \(-\sqrt{3}\):** \[ -\sqrt{3} \left( \left(x - \frac{7}{2}\right)^2 - \frac{13}{4} \right) = -\sqrt{3} \left(x - \frac{7}{2}\right)^2 + \frac{13 \sqrt{3}}{4} \] Thus, the completed square form of the quadratic expression is: \[ \boxed{-\sqrt{3} \left(x - \frac{7}{2}\right)^2 + \frac{13 \sqrt{3}}{4}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| deepseek-r1 | To complete the square for the quadratic \(-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}\): 1. Factor out \(-\sqrt{3}\) from the first two terms: \[ -\sqrt{3}(x^2 - 7x) - 9\sqrt{3} \] 2. Complete the square inside the parentheses. The coefficient of \(x\) is \(-7\). Half of \(-7\) is \(-\frac{7}{2}\), and squaring this gives \(\left(\frac{7}{2}\right)^2 = \frac{49}{4}\). Add and subtract \(\frac{49}{4}\) inside the parentheses: \[ -\sqrt{3}\left[(x^2 - 7x + \frac{49}{4}) - \frac{49}{4}\right] - 9\sqrt{3} \] 3. Simplify the expression: \[ -\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \sqrt{3} \cdot \frac{49}{4} - 9\sqrt{3} \] 4. Combine the constant terms. Convert \(9\sqrt{3}\) to a fraction with a denominator of 4: \[ \sqrt{3} \cdot \frac{49}{4} - 9\sqrt{3} = \frac{49}{4}\sqrt{3} - \frac{36}{4}\sqrt{3} = \frac{13}{4}\sqrt{3} \] 5. Write the final completed square form: \[ -\sqrt{3}\left(x - \frac{7}{2}\right)^2 + \frac{13}{4}\sqrt{3} \] The quadratic in completed square form is \(\boxed{ -\sqrt{3} \left(x - \dfrac{7}{2}\right)^2 + \dfrac{13}{4} \sqrt{3} }\). |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| gpto1 | To complete the square for the quadratic \( -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} \), we start by factoring out \(-\sqrt{3}\): \[ -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} = -\sqrt{3} (x^2 - 7 x + 9) \] Next, we complete the square inside the parentheses. For the quadratic \( x^2 - 7 x + 9 \), the steps are as follows: 1. Identify \( h \) such that \( x^2 - 7 x = (x - h)^2 - h^2 \). Here, \( h = \frac{7}{2} \). 2. Rewrite \( x^2 - 7 x + 9 \) as: \[ (x - \frac{7}{2})^2 - \left(\frac{7}{2}\right)^2 + 9 = (x - \frac{7}{2})^2 - \frac{49}{4} + 9 \] 3. Simplify the constants: \[ - \frac{49}{4} + 9 = - \frac{49}{4} + \frac{36}{4} = - \frac{13}{4} \] 4. The quadratic becomes: \[ (x - \frac{7}{2})^2 - \frac{13}{4} \] Now, plug this back into the expression and distribute \(-\sqrt{3}\): \[ - \sqrt{3} (x^2 - 7 x + 9) = -\sqrt{3}\left( (x - \frac{7}{2})^2 - \frac{13}{4} \right) = -\sqrt{3}(x - \frac{7}{2})^2 + \frac{13 \sqrt{3}}{4} \] Finally, the completed square form is: \[ \boxed{ -\sqrt{3}\left( x - \dfrac{7}{2} \right)^2 + \dfrac{13\sqrt{3}}{4} } \] \boxed{\, -\sqrt{3}\left( x-\dfrac{7}{2}\right)^{2}+\dfrac{13\sqrt{3}}{4}\,} |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| gpt-4o-2024-08-06 | To complete the square for the quadratic expression \(-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}\), we start by focusing on the quadratic and linear terms: \(-\sqrt{3} x^2 + 7 \sqrt{3} x\). First, factor out \(-\sqrt{3}\) from these two terms: \[ -\sqrt{3}(x^2 - \frac{7}{\sqrt{3}} x) \] Next, we complete the square inside the parentheses. The coefficient of \(x\) is \(-\frac{7}{\sqrt{3}}\). To complete the square, take half of this coefficient and square it: \[ \left(\frac{-\frac{7}{\sqrt{3}}}{2}\right)^2 = \left(\frac{-7}{2\sqrt{3}}\right)^2 = \frac{49}{12} \] Add and subtract this square inside the parentheses: \[ x^2 - \frac{7}{\sqrt{3}} x = \left(x^2 - \frac{7}{\sqrt{3}} x + \frac{49}{12}\right) - \frac{49}{12} \] This can be rewritten as: \[ \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{49}{12} \] Now, substitute back into the expression with the factor of \(-\sqrt{3}\): \[ -\sqrt{3} \left( \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{49}{12} \right) \] Distribute \(-\sqrt{3}\): \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 + \sqrt{3} \cdot \frac{49}{12} \] Simplify the constant term: \[ \sqrt{3} \cdot \frac{49}{12} = \frac{49\sqrt{3}}{12} \] Now, include the constant term from the original quadratic expression, \(-9\sqrt{3}\): \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 + \frac{49\sqrt{3}}{12} - 9\sqrt{3} \] Convert \(-9\sqrt{3}\) to a fraction with a denominator of 12: \[ -9\sqrt{3} = -\frac{108\sqrt{3}}{12} \] Combine the constant terms: \[ \frac{49\sqrt{3}}{12} - \frac{108\sqrt{3}}{12} = -\frac{59\sqrt{3}}{12} \] Thus, the completed square form of the quadratic is: \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{59\sqrt{3}}{12} \] The final answer is: \[ \boxed{-\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{59\sqrt{3}}{12}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| DeepSeek-Coder-V2-Lite-Instruct | To complete the square for the quadratic expression \(-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ -\sqrt{3} (x^2 - \frac{7}{\sqrt{3}} x) - 9 \sqrt{3} \] Simplify the coefficient inside the parentheses: \[ -\sqrt{3} (x^2 - \frac{7}{\sqrt{3}} x) - 9 \sqrt{3} = -\sqrt{3} (x^2 - \frac{7}{\sqrt{3}} x) - 9 \sqrt{3} \] \[ -\sqrt{3} (x^2 - \frac{7}{\sqrt{3}} x) - 9 \sqrt{3} = -\sqrt{3} (x^2 - \frac{7}{\sqrt{3}} x) - 9 \sqrt{3} \] 2. **Complete the square inside the parentheses:** To complete the square, we need to add and subtract the square of half the coefficient of \(x\): \[ x^2 - \frac{7}{\sqrt{3}} x \] The coefficient of \(x\) is \(-\frac{7}{\sqrt{3}}\), so half of this is \(-\frac{7}{2\sqrt{3}}\), and its square is: \[ \left(-\frac{7}{2\sqrt{3}}\right)^2 = \frac{49}{12} \] Add and subtract \(\frac{49}{12}\) inside the parentheses: \[ x^2 - \frac{7}{\sqrt{3}} x + \frac{49}{12} - \frac{49}{12} \] \[ x^2 - \frac{7}{\sqrt{3}} x + \frac{49}{12} - \frac{49}{12} = \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{49}{12} \] 3. **Substitute back into the expression:** \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 + \sqrt{3} \cdot \frac{49}{12} - 9 \sqrt{3} \] Simplify \(\sqrt{3} \cdot \frac{49}{12}\): \[ \sqrt{3} \cdot \frac{49}{12} = \frac{49 \sqrt{3}}{12} \] So the expression becomes: \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 + \frac{49 \sqrt{3}}{12} - 9 \sqrt{3} \] 4. **Combine like terms:** \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 + \frac{49 \sqrt{3}}{12} - 9 \sqrt{3} \] Simplify \(\frac{49 \sqrt{3}}{12} - 9 \sqrt{3}\): \[ \frac{49 \sqrt{3}}{12} - 9 \sqrt{3} = \frac{49 \sqrt{3} - 108 \sqrt{3}}{12} = \frac{-59 \sqrt{3}}{12} \] So the final expression is: \[ -\sqrt{3} \left(x - \frac{7}{2\sqrt{3}}\right)^2 - \frac{59 \sqrt{3}}{12} \] 5. **Simplify the expression inside the parentheses:** \[ x - \frac{7}{2\sqrt{3}} = x - \frac{7}{2\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = x - \frac{7 \sqrt{3}}{6} \] So the final expression is: \[ -\sqrt{3} \left(x - \frac{7 \sqrt{3}}{6}\right)^2 - \frac{59 \sqrt{3}}{12} \] Thus, the completed square form of the quadratic expression is: \[ \boxed{-\sqrt{3} \left(x - \frac{7 \sqrt{3}}{6}\right)^2 - \frac{59 \sqrt{3}}{12}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $-\sqrt{3} x^2+7 \sqrt{3} x-9 \sqrt{3}$. ## Step 2: Factor out the coefficient of $x^2$ from the quadratic equation To complete the square, we first need to ensure the coefficient of $x^2$ is 1. The coefficient of $x^2$ is $-\sqrt{3}$, so we factor this out: $-\sqrt{3}(x^2-7x+9)$. ## Step 3: Calculate the value to complete the square To complete the square for $x^2-7x$, we take half of the coefficient of $x$, which is $-\frac{7}{2}$, and then square it: $\left(-\frac{7}{2}\right)^2 = \frac{49}{4}$. ## Step 4: Add and subtract the calculated value within the parentheses Adding and subtracting $\frac{49}{4}$ inside the parentheses allows us to create a perfect square trinomial without changing the value of the expression: $-\sqrt{3}\left(x^2-7x+\frac{49}{4}-\frac{49}{4}+9\right)$. ## Step 5: Rewrite the expression with the completed square Group the terms that form the perfect square trinomial: $-\sqrt{3}\left(\left(x^2-7x+\frac{49}{4}\right)-\frac{49}{4}+9\right)$. This simplifies to $-\sqrt{3}\left(\left(x-\frac{7}{2}\right)^2-\frac{49}{4}+9\right)$. ## Step 6: Simplify the constants inside the parentheses Combine the constants: $-\frac{49}{4}+9 = -\frac{49}{4}+\frac{36}{4} = -\frac{13}{4}$. ## Step 7: Rewrite the final expression Thus, the expression simplifies to $-\sqrt{3}\left(\left(x-\frac{7}{2}\right)^2-\frac{13}{4}\right)$. ## Step 8: Distribute $-\sqrt{3}$ for final form Distribute $-\sqrt{3}$ across the terms inside the parentheses for the final form: $-\sqrt{3}\left(x-\frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}$. The final answer is: $\boxed{-\sqrt{3}\left(x-\frac{7}{2}\right)^2 + \frac{13\sqrt{3}}{4}}$ |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $-\sqrt{3} x^2+7 \sqrt{3} x-9 \sqrt{3}$. ## Step 2: Factor out the coefficient of $x^2$ from the quadratic and linear terms We factor out $-\sqrt{3}$ from the first two terms to get $-\sqrt{3}(x^2-7x)-9\sqrt{3}$. ## Step 3: Calculate the value to complete the square To complete the square for $x^2-7x$, we need to add and subtract $(\frac{7}{2})^2 = \frac{49}{4}$ inside the parenthesis. ## Step 4: Add and subtract the calculated value Adding and subtracting $\frac{49}{4}$ inside the parenthesis, we get $-\sqrt{3}(x^2-7x+\frac{49}{4}-\frac{49}{4})-9\sqrt{3}$. ## Step 5: Rewrite the expression with the completed square This can be rewritten as $-\sqrt{3}((x-\frac{7}{2})^2-\frac{49}{4})-9\sqrt{3}$. ## Step 6: Distribute the $-\sqrt{3}$ and simplify Distributing the $-\sqrt{3}$ gives $-\sqrt{3}(x-\frac{7}{2})^2+\frac{49\sqrt{3}}{4}-9\sqrt{3}$. ## Step 7: Combine like terms Combining the constants gives $-\sqrt{3}(x-\frac{7}{2})^2+\frac{49\sqrt{3}-36\sqrt{3}}{4}$. ## Step 8: Simplify the constants Simplifying the constants gives $-\sqrt{3}(x-\frac{7}{2})^2+\frac{13\sqrt{3}}{4}$. The final answer is: $\boxed{-\sqrt{3}(x-\frac{7}{2})^2+\frac{13\sqrt{3}}{4}}$ |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| Qwen2-72B-Instruct | To complete the square for the given quadratic expression, we first factor out the coefficient of \(x^2\), which is \(-\sqrt{3}\). This gives us: \[ -\sqrt{3}(x^2 - 7x + 9) \] Next, we focus on completing the square inside the parentheses. To do this, we take half of the coefficient of \(x\) (which is \(-7\)), square it, and add it to the expression. We also subtract it to keep the equation balanced. Half of \(-7\) is \(-3.5\), and squaring that gives us \(12.25\). So, we rewrite the expression as: \[ -\sqrt{3}(x^2 - 7x + 12.25 - 12.25 + 9) \] This can be rearranged as: \[ -\sqrt{3}[(x^2 - 7x + 12.25) - 12.25 + 9] \] The term inside the first set of brackets is now a perfect square trinomial, which can be written as: \[ (x - 3.5)^2 \] Substituting this back into our expression, we get: \[ -\sqrt{3}[(x - 3.5)^2 - 12.25 + 9] \] Simplifying the constants outside the square: \[ -\sqrt{3}[(x - 3.5)^2 - 3.25] \] Finally, distributing the \(-\sqrt{3}\): \[ -\sqrt{3}(x - 3.5)^2 + 3.25\sqrt{3} \] Therefore, the completed square form of the given quadratic is: \[ \boxed{-\sqrt{3}(x - 3.5)^2 + 3.25\sqrt{3}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| baichuan4 | To complete the square for the quadratic $-\sqrt{3} x^2+7 \sqrt{3} x-9 \sqrt{3}$ we first need to isolate the quadratic term and the linear term. We can do this by factoring out the coefficient of the quadratic term, which is $-\sqrt{3}$. We get: $$-\sqrt{3}(x^2 - 7x + 9)$$ Next, we need to complete the square inside the parentheses. To do this, we look at the linear term, $-7x$, and take half of its coefficient, which is $-7/2$. Then we square this number to get $(-7/2)^2 = 49/4$. This is the number we need to add and subtract inside the parentheses to complete the square. However, we also need to adjust the constant term inside the parentheses. Since we added $49/4$, we need to subtract the same amount to keep the equation balanced. The original constant term was $+9$, so we subtract $49/4$ from this to get $9 - 49/4 = 36/4 - 49/4 = -13/4$. Now we can write the completed square as: $$-\sqrt{3}(x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 9)$$ Simplifying this, we get: $$-\sqrt{3}((x - \frac{7}{2})^2 - \frac{49}{4} + \frac{36}{4})$$ Further simplifying: $$-\sqrt{3}((x - \frac{7}{2})^2 - \frac{13}{4})$$ Finally, we can distribute the $-\sqrt{3}$ back into the parentheses: $$-\sqrt{3}(x - \frac{7}{2})^2 + \sqrt{3} \cdot \frac{13}{4}$$ Simplifying the last term: $$-\sqrt{3}(x - \frac{7}{2})^2 + \frac{13\sqrt{3}}{4}$$ This is the completed square form of the original quadratic. The final answer is: $$\boxed{-\sqrt{3}(x - \frac{7}{2})^2 + \frac{13\sqrt{3}}{4}}$$ |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| doubao | \[ \begin{align*} &-\sqrt{3}x^2+7\sqrt{3}x-9\sqrt{3}\\ =&-\sqrt{3}(x^2-7x)-9\sqrt{3}\\ =&-\sqrt{3}(x^2-7x+\frac{49}{4}-\frac{49}{4})-9\sqrt{3}\\ =&-\sqrt{3}[(x-\frac{7}{2})^2-\frac{49}{4}]-9\sqrt{3}\\ =&-\sqrt{3}(x-\frac{7}{2})^2+\frac{49\sqrt{3}}{4}-9\sqrt{3}\\ =&-\sqrt{3}(x-\frac{7}{2})^2+\frac{49\sqrt{3}-36\sqrt{3}}{4}\\ =&-\sqrt{3}(x-\frac{7}{2})^2+\frac{13\sqrt{3}}{4} \end{align*} \] So the answer is $\boxed{-\sqrt{3}(x-\frac{7}{2})^2+\frac{13\sqrt{3}}{4}}$. |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| gpto1 | To complete the square for the quadratic \( -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} \), we start by factoring out \(-\sqrt{3}\): \[ -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} = -\sqrt{3} (x^2 - 7 x + 9) \] Next, we complete the square inside the parentheses. For the quadratic \( x^2 - 7 x + 9 \), the steps are as follows: 1. Identify \( h \) such that \( x^2 - 7 x = (x - h)^2 - h^2 \). Here, \( h = \frac{7}{2} \). 2. Rewrite \( x^2 - 7 x + 9 \) as: \[ (x - \frac{7}{2})^2 - \left(\frac{7}{2}\right)^2 + 9 = (x - \frac{7}{2})^2 - \frac{49}{4} + 9 \] 3. Simplify the constants: \[ - \frac{49}{4} + 9 = - \frac{49}{4} + \frac{36}{4} = - \frac{13}{4} \] 4. The quadratic becomes: \[ (x - \frac{7}{2})^2 - \frac{13}{4} \] Now, plug this back into the expression and distribute \(-\sqrt{3}\): \[ - \sqrt{3} (x^2 - 7 x + 9) = -\sqrt{3}\left( (x - \frac{7}{2})^2 - \frac{13}{4} \right) = -\sqrt{3}(x - \frac{7}{2})^2 + \frac{13 \sqrt{3}}{4} \] Finally, the completed square form is: \[ \boxed{ -\sqrt{3}\left( x - \dfrac{7}{2} \right)^2 + \dfrac{13\sqrt{3}}{4} } \] \boxed{\, -\sqrt{3}\left( x-\dfrac{7}{2}\right)^{2}+\dfrac{13\sqrt{3}}{4}\,} |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| kimi | To complete the square for the quadratic expression \(-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} = -\sqrt{3} (x^2 - 7x) - 9 \sqrt{3} \] 2. **Complete the square inside the parentheses:** To complete the square for \(x^2 - 7x\), we need to add and subtract the square of half the coefficient of \(x\). The coefficient of \(x\) is \(-7\), so half of it is \(-\frac{7}{2}\), and squaring it gives \(\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\). \[ x^2 - 7x = \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} \] Substituting this back into the expression, we get: \[ -\sqrt{3} \left( \left(x - \frac{7}{2}\right)^2 - \frac{49}{4} \right) - 9 \sqrt{3} \] 3. **Distribute \(-\sqrt{3}\) through the parentheses:** \[ -\sqrt{3} \left(x - \frac{7}{2}\right)^2 + \frac{49 \sqrt{3}}{4} - 9 \sqrt{3} \] 4. **Combine the constant terms:** To combine \(\frac{49 \sqrt{3}}{4}\) and \(-9 \sqrt{3}\), we need a common denominator. The common denominator is 4, so we rewrite \(-9 \sqrt{3}\) as \(-\frac{36 \sqrt{3}}{4}\). \[ \frac{49 \sqrt{3}}{4} - \frac{36 \sqrt{3}}{4} = \frac{13 \sqrt{3}}{4} \] So the expression becomes: \[ -\sqrt{3} \left(x - \frac{7}{2}\right)^2 + \frac{13 \sqrt{3}}{4} \] 5. **Write the final answer in a box:** \[ \boxed{-\sqrt{3} \left(x - \frac{7}{2}\right)^2 + \frac{13 \sqrt{3}}{4}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| minimax | To complete the square for the quadratic equation $-\sqrt{3} x^2 + 7\sqrt{3} x - 9\sqrt{3}$, we need to express it in the form $a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. First, let's factor out the coefficient of $x^2$ from the first two terms: $$-\sqrt{3}(x^2 - 7x) - 9\sqrt{3}$$ Now, to complete the square, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses. The coefficient of $x$ is $-7$, so half of that is $-\frac{7}{2}$, and squaring that gives us $\left(-\frac{7}{2}\right)^2 = \frac{49}{4}$. We add and subtract $\frac{49}{4}$ inside the parentheses, but since we factored out $-\sqrt{3}$, we need to add and subtract $-\sqrt{3} \cdot \frac{49}{4}$ outside the parentheses to keep the equation balanced: $$-\sqrt{ |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| wenxinyiyan | 首先，我们可以将给定的二次式写为： $-\sqrt{3} x^2+7 \sqrt{3} x-9 \sqrt{3} = -\sqrt{3}(x^2 - 7x + 9)$ 为了完成平方，我们需要使二次项和一次项形成一个完全平方项。我们知道$(x-a)^2 = x^2 - 2ax + a^2$，所以我们需要找到$a$使得$2a = 7$，从而得到$a = \frac{7}{2}$。 接下来，我们可以将$a^2$即$\left(\frac{7}{2}\right)^2 = \frac{49}{4}$加到式子中并同时减去相同的量以保持等式的平衡： $-\sqrt{3}(x^2 - 7x + 9) = -\sqrt{3}\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 9\right)$ 现在我们可以将前三项组合成一个完全平方项： $-\sqrt{3}\left(x^2 - 7x + \frac{49}{4} - \frac{49}{4} + 9\right) = -\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 9\right)$ 最后，我们可以将常数项合并： $-\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{49}{4} + 9\right) = -\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{13}{4}\right)$ 所以，完成平方后的二次式为： $\boxed{-\sqrt{3}\left(\left(x - \frac{7}{2}\right)^2 - \frac{13}{4}\right)}$ |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| zhipu | To complete the square for the quadratic $-\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3}$, we follow these steps: 1. **Factor out the common term** $-\sqrt{3}$ from the first two terms: \[ -\sqrt{3} x^2 + 7 \sqrt{3} x - 9 \sqrt{3} = -\sqrt{3} (x^2 - 7x) - 9 \sqrt{3} \] 2. **Complete the square** inside the parentheses: - Take half of the coefficient of $x$, which is $7$, and square it: \[ \left(\frac{7}{2}\right)^2 = \frac{49}{4} \] - Add and subtract this square inside the parentheses: \[ -\sqrt{3} (x^2 - 7x + \frac{49}{4} - \frac{49}{4}) - 9 \sqrt{3} \] - This can be rewritten as: \[ -\sqrt{3} \left((x - \frac{7}{2})^2 - \frac{49}{4}\right) - 9 \sqrt{3} \] 3. **Distribute** the $-\sqrt{3}$ and simplify: \[ -\sqrt{3} (x - \frac{7}{2})^2 + \sqrt{3} \cdot \frac{49}{4} - 9 \sqrt{3} \] \[ -\sqrt{3} (x - \frac{7}{2})^2 + \frac{49 \sqrt{3}}{4} - 9 \sqrt{3} \] 4. **Combine the constant terms**: - Convert $9 \sqrt{3}$ to a fraction with denominator 4: \[ 9 \sqrt{3} = \frac{36 \sqrt{3}}{4} \] - Now combine the constants: \[ \frac{49 \sqrt{3}}{4} - \frac{36 \sqrt{3}}{4} = \frac{13 \sqrt{3}}{4} \] 5. **Write the final expression**: \[ -\sqrt{3} (x - \frac{7}{2})^2 + \frac{13 \sqrt{3}}{4} \] Thus, the completed square form of the quadratic is: \[ \boxed{-\sqrt{3} (x - \frac{7}{2})^2 + \frac{13 \sqrt{3}}{4}} \] |
\frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-0552fcc6-bebf-9716-9943-ded5e91cb579', 'request_id': '0552fcc6-bebf-9716-9943-ded5e91cb579'} | \frac{13 \sqrt{3}}{4}-\sqrt{3} \left(x-\frac{7}{2}\right)^2 |