| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To complete the square for the quadratic expression \(-9x^2 + 5x + \frac{5}{2}\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ -9x^2 + 5x + \frac{5}{2} = -9 \left( x^2 - \frac{5}{9}x \right) + \frac{5}{2} \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\) inside the parentheses, which is \(-\frac{5}{9}\), divide it by 2, and square it: \[ \left( -\frac{5}{9} \div 2 \right)^2 = \left( -\frac{5}{18} \right)^2 = \frac{25}{324} \] - Add and subtract this square inside the parentheses: \[ -9 \left( x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324} \right) + \frac{5}{2} \] - Simplify the expression inside the parentheses: \[ -9 \left( \left( x - \frac{5}{18} \right)^2 - \frac{25}{324} \right) + \frac{5}{2} \] 3. **Distribute the \(-9\) and simplify:** \[ -9 \left( x - \frac{5}{18} \right)^2 + 9 \cdot \frac{25}{324} + \frac{5}{2} \] - Calculate \(9 \cdot \frac{25}{324}\): \[ 9 \cdot \frac{25}{324} = \frac{225}{324} = \frac{25}{36} \] - Add \(\frac{25}{36}\) to \(\frac{5}{2}\): \[ \frac{5}{2} = \frac{90}{36} \] \[ \frac{90}{36} + \frac{25}{36} = \frac{115}{36} \] 4. **Combine all parts:** \[ -9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36} \] Thus, the completed square form of the quadratic expression is: \[ \boxed{-9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| deepseek-r1 | To complete the square for the quadratic \(-9x^2 + 5x + \frac{5}{2}\), we start by factoring out the coefficient of \(x^2\) from the first two terms: \[ -9x^2 + 5x + \frac{5}{2} = -9\left(x^2 - \frac{5}{9}x\right) + \frac{5}{2} \] Next, we complete the square inside the parentheses. The coefficient of \(x\) is \(-\frac{5}{9}\). We take half of this coefficient, which is \(-\frac{5}{18}\), and square it to get \(\left(-\frac{5}{18}\right)^2 = \frac{25}{324}\). We add and subtract this value inside the parentheses: \[ -9\left(x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324}\right) + \frac{5}{2} \] This can be rewritten as: \[ -9\left(\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right) + \frac{5}{2} \] Distributing the \(-9\) and combining the constant terms: \[ -9\left(x - \frac{5}{18}\right)^2 + 9 \cdot \frac{25}{324} + \frac{5}{2} \] Simplifying the constants: \[ 9 \cdot \frac{25}{324} = \frac{225}{324} = \frac{25}{36} \] Converting \(\frac{5}{2}\) to a fraction with a denominator of 36: \[ \frac{5}{2} = \frac{90}{36} \] Adding the constants: \[ \frac{25}{36} + \frac{90}{36} = \frac{115}{36} \] Thus, the quadratic in completed square form is: \[ \boxed{-9\left(x - \dfrac{5}{18}\right)^2 + \dfrac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| gpto1 | To complete the square for the quadratic expression \( -9x^2 + 5x + \dfrac{5}{2} \), we'll follow these steps: 1. **Factor out the coefficient of \( x^2 \) from the first two terms:** \[ -9x^2 + 5x = -9\left(x^2 - \dfrac{5}{9}x\right) \] 2. **Complete the square inside the parentheses:** - Find the value to complete the square: \[ \left(\dfrac{-\dfrac{5}{9}}{2}\right)^2 = \left(-\dfrac{5}{18}\right)^2 = \dfrac{25}{324} \] - Add and subtract this value inside the parentheses: \[ x^2 - \dfrac{5}{9}x = \left(x - \dfrac{5}{18}\right)^2 - \dfrac{25}{324} \] 3. **Rewrite the original expression with the completed square:** \[ -9\left(x^2 - \dfrac{5}{9}x\right) + \dfrac{5}{2} = -9\left[\left(x - \dfrac{5}{18}\right)^2 - \dfrac{25}{324}\right] + \dfrac{5}{2} \] 4. **Simplify the expression:** - Distribute the \(-9\): \[ -9\left(x - \dfrac{5}{18}\right)^2 + \dfrac{225}{324} + \dfrac{5}{2} \] - Simplify constants: \[ \dfrac{225}{324} = \dfrac{25}{36} \] \[ \dfrac{5}{2} = \dfrac{90}{36} \] \[ \dfrac{25}{36} + \dfrac{90}{36} = \dfrac{115}{36} \] - Final simplified form: \[ -9\left(x - \dfrac{5}{18}\right)^2 + \dfrac{115}{36} \] So, the quadratic expression in completed square form is: \[ \boxed{\, -9\left(x - \dfrac{5}{18}\right)^{2} + \dfrac{115}{36} \,} \] \boxed{\, -9\left(x-\dfrac{5}{18}\right)^{2}+\dfrac{115}{36} \,} |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| gpt-4o-2024-08-06 | To complete the square for the quadratic expression \(-9x^2 + 5x + \frac{5}{2}\), we start by focusing on the quadratic and linear terms, \(-9x^2 + 5x\). First, factor out \(-9\) from the quadratic and linear terms: \[ -9x^2 + 5x = -9(x^2 - \frac{5}{9}x) \] Next, we complete the square for the expression inside the parentheses, \(x^2 - \frac{5}{9}x\). To do this, take half of the coefficient of \(x\), which is \(-\frac{5}{9}\), and square it: \[ \left(\frac{-\frac{5}{9}}{2}\right)^2 = \left(-\frac{5}{18}\right)^2 = \frac{25}{324} \] Add and subtract this square inside the parentheses: \[ x^2 - \frac{5}{9}x = \left(x - \frac{5}{18}\right)^2 - \frac{25}{324} \] Substitute back into the expression with the factor of \(-9\): \[ -9(x^2 - \frac{5}{9}x) = -9\left(\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right) \] Distribute the \(-9\): \[ = -9\left(x - \frac{5}{18}\right)^2 + \frac{225}{324} \] Simplify \(\frac{225}{324}\): \[ \frac{225}{324} = \frac{25}{36} \] Now, add the constant term \(\frac{5}{2}\) back to the expression: \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{5}{2} \] Convert \(\frac{5}{2}\) to a fraction with a denominator of 36: \[ \frac{5}{2} = \frac{90}{36} \] Add the fractions: \[ \frac{25}{36} + \frac{90}{36} = \frac{115}{36} \] Thus, the completed square form of the quadratic is: \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36} \] The final answer is: \[ \boxed{-9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| DeepSeek-Coder-V2-Lite-Instruct | To complete the square for the quadratic expression \(-9 x^2 + 5 x + \frac{5}{2}\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ -9 x^2 + 5 x + \frac{5}{2} = -9 \left( x^2 - \frac{5}{9} x \right) + \frac{5}{2} \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\) inside the parentheses, which is \(-\frac{5}{9}\). - Divide it by 2: \(\frac{-\frac{5}{9}}{2} = -\frac{5}{18}\). - Square this result: \(\left( -\frac{5}{18} \right)^2 = \frac{25}{324}\). 3. **Add and subtract this square inside the parentheses:** \[ -9 \left( x^2 - \frac{5}{9} x + \frac{25}{324} - \frac{25}{324} \right) + \frac{5}{2} \] \[ = -9 \left( x^2 - \frac{5}{9} x + \frac{25}{324} \right) + 9 \cdot \frac{25}{324} + \frac{5}{2} \] \[ = -9 \left( x - \frac{5}{18} \right)^2 + \frac{225}{324} + \frac{5}{2} \] 4. **Simplify the constants:** - Convert \(\frac{225}{324}\) to a fraction with a common denominator of 324: \[ \frac{225}{324} = \frac{225}{324} = \frac{225}{324} = \frac{25}{36} \] - Convert \(\frac{5}{2}\) to a fraction with a common denominator of 324: \[ \frac{5}{2} = \frac{5 \times 162}{2 \times 162} = \frac{810}{324} \] - Add the fractions: \[ \frac{225}{324} + \frac{810}{324} = \frac{1035}{324} = \frac{115}{36} \] 5. **Combine everything:** \[ -9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36} \] Thus, the completed square form of the quadratic expression is: \[ \boxed{-9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $-9x^2 + 5x + \frac{5}{2}$. ## Step 2: Determine the coefficient of $x^2$ The coefficient of $x^2$ is $-9$. ## Step 3: Calculate the value to complete the square To complete the square, we need to add and subtract $(\frac{b}{2a})^2$ inside the parentheses, where $a$ is the coefficient of $x^2$ and $b$ is the coefficient of $x$. In this case, $a = -9$ and $b = 5$. So, the value to complete the square is $(\frac{5}{2(-9)})^2 = (\frac{5}{-18})^2 = (\frac{-5}{18})^2 = \frac{25}{324}$. ## Step 4: Rewrite the quadratic equation in completed square form First, factor out the coefficient of $x^2$ from the quadratic and linear terms: $-9(x^2 - \frac{5}{9}x) + \frac{5}{2}$. Then, add and subtract the value calculated in step 3: $-9(x^2 - \frac{5}{9}x + \frac{25}{324}) + \frac{5}{2} + \frac{9 \cdot 25}{324}$. Simplify the expression: $-9(x - \frac{5}{18})^2 + \frac{5}{2} + \frac{25}{36}$. ## Step 5: Combine the constant terms Combine the constant terms: $\frac{5}{2} + \frac{25}{36} = \frac{90}{36} + \frac{25}{36} = \frac{115}{36}$. ## Step 6: Write the final completed square form The final completed square form is $-9(x - \frac{5}{18})^2 + \frac{115}{36}$. The final answer is: $\boxed{-9(x - \frac{5}{18})^2 + \frac{115}{36}}$ |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the given quadratic equation The given quadratic equation is $-9x^2 + 5x + \frac{5}{2}$. ## Step 2: Factor out the coefficient of $x^2$ from the first two terms To complete the square, we first factor out the coefficient of $x^2$, which is $-9$, from the first two terms: $-9x^2 + 5x = -9(x^2 - \frac{5}{9}x)$. ## Step 3: Find the value to complete the square To complete the square for $x^2 - \frac{5}{9}x$, we need to add $(\frac{1}{2} \times -\frac{5}{9})^2 = (\frac{-5}{18})^2 = \frac{25}{324}$ inside the parenthesis. ## Step 4: Add and subtract the value to complete the square Adding and subtracting $\frac{25}{324}$ inside the parenthesis: $-9(x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324})$. ## Step 5: Rewrite the expression with the completed square Rewrite the expression with the completed square: $-9(x^2 - \frac{5}{9}x + \frac{25}{324}) + 9 \times \frac{25}{324} + \frac{5}{2}$. ## Step 6: Simplify the expression Simplify the expression: $-9(x - \frac{5}{18})^2 + \frac{25}{36} + \frac{5}{2}$. ## Step 7: Combine the constant terms Combine the constant terms: $-9(x - \frac{5}{18})^2 + \frac{25}{36} + \frac{90}{36}$. ## Step 8: Simplify the constant terms Simplify the constant terms: $-9(x - \frac{5}{18})^2 + \frac{115}{36}$. The final answer is: $\boxed{-9(x - \frac{5}{18})^2 + \frac{115}{36}}$ |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| Qwen2-72B-Instruct | To complete the square for the quadratic expression \(-9x^2 + 5x + \frac{5}{2}\), we first factor out the coefficient of \(x^2\) from the first two terms: \[ -9x^2 + 5x = -9(x^2 - \frac{5}{9}x) \] Next, we find the value that completes the square. This is done by taking half of the coefficient of \(x\) inside the parentheses and squaring it. The coefficient of \(x\) inside the parentheses is \(-\frac{5}{9}\), so half of this is \(-\frac{5}{18}\). Squaring this gives us \(\left(-\frac{5}{18}\right)^2 = \frac{25}{324}\). We add and subtract this value inside the parentheses to complete the square: \[ -9\left(x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324}\right) = -9\left[\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right] \] Simplifying further: \[ -9\left[\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right] = -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} \] Now, we need to account for the constant term \(\frac{5}{2}\) in the original expression. Adding this to our completed square form: \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{5}{2} \] Since \(\frac{5}{2} = \frac{90}{36}\), we can combine the constants: \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{90}{36} = -9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36} \] Thus, the completed square form of the given quadratic expression is: \[ \boxed{-9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| baichuan4 | To complete the square for the quadratic $-9x^2 + 5x + \frac{5}{2}$, we first need to identify the coefficients of the quadratic terms and the linear term. We have $a = -9$, $b = 5$, and $c = \frac{5}{2}$. The goal is to rewrite the quadratic in the form $(ax + b)^2 + c$. To do this, we need to find the value of $b$ that will make the expression a perfect square. This value is given by $b = \frac{-a}{2}$. For our quadratic, we have $a = -9$, so $b$ should be $\frac{-(-9)}{2} = \frac{9}{2}$. However, our quadratic has a linear term coefficient of $5$, which is not equal to $2 \cdot \frac{9}{2} = 9$. Therefore, we need to adjust the expression to include the correct value of $b$. We start by factoring out the $a$ coefficient from the quadratic and linear terms: $$-9x^2 + 5x + \frac{5}{2} = -9\left(x^2 - \frac{5}{9}x\right) + \frac{5}{2}.$$ Next, we add and subtract the square of $\frac{b}{2}$ to complete the square. Since $b = \frac{9}{2}$, the square of $\frac{b}{2}$ is $\left(\frac{9}{4}\right)^2 = \frac{81}{16}$. We add this inside the parentheses and also subtract it outside to keep the expression balanced: $$-9\left(x^2 - \frac{5}{9}x + \frac{81}{16}\right) - \frac{81}{16} \cdot 9 + \frac{5}{2}.$$ Now, we can rewrite the expression inside the parentheses as a perfect square: $$-9\left(\left(x - \frac{9}{4}\right)^2\right) - \frac{81}{16} \cdot 9 + \frac{5}{2}.$$ Simplifying the constant terms, we get: $$-9\left(x - \frac{9}{4}\right)^2 - \frac{729}{16} + \frac{5}{2}.$$ Converting $\frac{5}{2}$ to a fraction with denominator $16$, we have $\frac{5}{2} = \frac{40}{16}$. Adding this to the other constant terms, we get: $$-9\left(x - \frac{9}{4}\right)^2 - \frac{729}{16} + \frac{40}{16}.$$ Simplifying the constant terms further, we get: $$-9\left(x - \frac{9}{4}\right)^2 - \frac{689}{16}.$$ Finally, we can rewrite the entire expression in a more standard form: $$\boxed{-9\left(x - \frac{9}{4}\right)^2 + \frac{689}{16}}.$$ This is the completed square form of the original quadratic. |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| doubao | \[ \begin{align*} -9x^2+5x+\frac{5}{2}&=-9(x^2-\frac{5}{9}x)+ \frac{5}{2}\\ &=-9(x^2-\frac{5}{9}x+\frac{25}{324}-\frac{25}{324})+ \frac{5}{2}\\ &=-9(x^2-\frac{5}{9}x+\frac{25}{324})+(-9)\times(-\frac{25}{324})+ \frac{5}{2}\\ &=-9(x-\frac{5}{18})^2+\frac{25}{36}+\frac{5}{2}\\ &=-9(x-\frac{5}{18})^2+\frac{25}{36}+\frac{90}{36}\\ &=-9(x-\frac{5}{18})^2+\frac{115}{36}\\ \end{align*} \] So the answer is $\boxed{-9(x-\frac{5}{18})^2+\frac{115}{36}}$. |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| gpto1 | To complete the square for the quadratic expression \( -9x^2 + 5x + \dfrac{5}{2} \), we'll follow these steps: 1. **Factor out the coefficient of \( x^2 \) from the first two terms:** \[ -9x^2 + 5x = -9\left(x^2 - \dfrac{5}{9}x\right) \] 2. **Complete the square inside the parentheses:** - Find the value to complete the square: \[ \left(\dfrac{-\dfrac{5}{9}}{2}\right)^2 = \left(-\dfrac{5}{18}\right)^2 = \dfrac{25}{324} \] - Add and subtract this value inside the parentheses: \[ x^2 - \dfrac{5}{9}x = \left(x - \dfrac{5}{18}\right)^2 - \dfrac{25}{324} \] 3. **Rewrite the original expression with the completed square:** \[ -9\left(x^2 - \dfrac{5}{9}x\right) + \dfrac{5}{2} = -9\left[\left(x - \dfrac{5}{18}\right)^2 - \dfrac{25}{324}\right] + \dfrac{5}{2} \] 4. **Simplify the expression:** - Distribute the \(-9\): \[ -9\left(x - \dfrac{5}{18}\right)^2 + \dfrac{225}{324} + \dfrac{5}{2} \] - Simplify constants: \[ \dfrac{225}{324} = \dfrac{25}{36} \] \[ \dfrac{5}{2} = \dfrac{90}{36} \] \[ \dfrac{25}{36} + \dfrac{90}{36} = \dfrac{115}{36} \] - Final simplified form: \[ -9\left(x - \dfrac{5}{18}\right)^2 + \dfrac{115}{36} \] So, the quadratic expression in completed square form is: \[ \boxed{\, -9\left(x - \dfrac{5}{18}\right)^{2} + \dfrac{115}{36} \,} \] \boxed{\, -9\left(x-\dfrac{5}{18}\right)^{2}+\dfrac{115}{36} \,} |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| kimi | To complete the square for the quadratic expression \(-9 x^2 + 5 x + \frac{5}{2}\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ -9 x^2 + 5 x + \frac{5}{2} = -9 \left( x^2 - \frac{5}{9} x \right) + \frac{5}{2} \] 2. **Complete the square inside the parentheses:** To complete the square for \(x^2 - \frac{5}{9} x\), we need to add and subtract the square of half the coefficient of \(x\). The coefficient of \(x\) is \(-\frac{5}{9}\), so half of it is \(-\frac{5}{18}\), and squaring it gives \(\left( -\frac{5}{18} \right)^2 = \frac{25}{324}\). \[ x^2 - \frac{5}{9} x = \left( x - \frac{5}{18} \right)^2 - \frac{25}{324} \] Substituting this back into the expression, we get: \[ -9 \left( x^2 - \frac{5}{9} x \right) + \frac{5}{2} = -9 \left( \left( x - \frac{5}{18} \right)^2 - \frac{25}{324} \right) + \frac{5}{2} \] 3. **Distribute the \(-9\) through the parentheses:** \[ -9 \left( \left( x - \frac{5}{18} \right)^2 - \frac{25}{324} \right) = -9 \left( x - \frac{5}{18} \right)^2 + \frac{225}{324} \] Simplify \(\frac{225}{324}\): \[ \frac{225}{324} = \frac{25}{36} \] So the expression becomes: \[ -9 \left( x - \frac{5}{18} \right)^2 + \frac{25}{36} + \frac{5}{2} \] 4. **Combine the constant terms:** To combine \(\frac{25}{36}\) and \(\frac{5}{2}\), we need a common denominator. The common denominator is 36. \[ \frac{5}{2} = \frac{90}{36} \] So: \[ \frac{25}{36} + \frac{90}{36} = \frac{115}{36} \] Therefore, the expression is: \[ -9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36} \] 5. **Write the final answer in a box:** \[ \boxed{-9 \left( x - \frac{5}{18} \right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| minimax | To complete the square for the quadratic $-9x^2 + 5x + \frac{5}{2}$, 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: $$-9(x^2 - \frac{5}{9}x) + \frac{5}{2}$$ Now, to complete the square inside the parentheses, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses. The coefficient of $x$ inside the parentheses is $-\frac{5}{9}$, so half of that is $-\frac{5}{18}$. Squaring $-\frac{5}{18}$ gives us $\left(-\frac{5}{18}\right)^2 = \frac{25}{324}$. We add and subtract this value inside the parentheses: $$-9\left(x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324}\ |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| wenxinyiyan | 首先，将给定的二次式写为： $-9 x^2+5 x+\frac{5}{2} = -9\left(x^2 - \frac{5}{9}x\right) + \frac{5}{2}$ 为了完成平方，我们需要加上和减去 $\left(\frac{5}{18}\right)^2$： $= -9\left(x^2 - \frac{5}{9}x + \left(\frac{5}{18}\right)^2 - \left(\frac{5}{18}\right)^2\right) + \frac{5}{2}$ $= -9\left(x - \frac{5}{18}\right)^2 + 9\left(\frac{5}{18}\right)^2 + \frac{5}{2}$ $= -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{5}{2}$ $= -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{90}{36}$ $= -9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}$ $= -9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}$ 所以，完成平方后的表达式为： $\boxed{-9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}}$ |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| zhipu | To complete the square for the quadratic $-9x^2 + 5x + \frac{5}{2}$, we follow these steps: 1. **Factor out the coefficient of $x^2$ from the first two terms:** \[ -9x^2 + 5x + \frac{5}{2} = -9(x^2 - \frac{5}{9}x) + \frac{5}{2} \] 2. **Complete the square inside the parentheses:** - Take half of the coefficient of $x$, square it, and add and subtract it inside the parentheses: \[ -9(x^2 - \frac{5}{9}x) = -9\left(x^2 - \frac{5}{9}x + \left(\frac{5}{18}\right)^2 - \left(\frac{5}{18}\right)^2\right) \] - Simplify the squared term: \[ \left(\frac{5}{18}\right)^2 = \frac{25}{324} \] - So, we have: \[ -9\left(x^2 - \frac{5}{9}x + \frac{25}{324} - \frac{25}{324}\right) = -9\left(\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right) \] 3. **Distribute the $-9$ and simplify:** \[ -9\left(\left(x - \frac{5}{18}\right)^2 - \frac{25}{324}\right) = -9\left(x - \frac{5}{18}\right)^2 + 9 \cdot \frac{25}{324} \] - Simplify the constant term: \[ 9 \cdot \frac{25}{324} = \frac{225}{324} = \frac{25}{36} \] 4. **Combine the constant terms:** \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{25}{36} + \frac{5}{2} \] - Convert $\frac{5}{2}$ to a common denominator with $\frac{25}{36}$: \[ \frac{5}{2} = \frac{90}{36} \] - Add the fractions: \[ \frac{25}{36} + \frac{90}{36} = \frac{115}{36} \] Thus, the completed square form of the quadratic is: \[ -9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36} \] Therefore, the final answer is: \[ \boxed{-9\left(x - \frac{5}{18}\right)^2 + \frac{115}{36}} \] |
\frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-ec4f5ecd-dc64-9502-a394-4bc04ff62115', 'request_id': 'ec4f5ecd-dc64-9502-a394-4bc04ff62115'} | \frac{115}{36}-9 \left(x-\frac{5}{18}\right)^2 |