| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 5(x^2 - \frac{8}{5}x) - 16 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-\frac{8}{5}\), divide it by 2, and square it: \[ \left(\frac{-\frac{8}{5}}{2}\right)^2 = \left(\frac{-8}{10}\right)^2 = \left(\frac{-4}{5}\right)^2 = \frac{16}{25} \] - Add and subtract this square inside the parentheses: \[ 5\left(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}\right) - 16 \] - Simplify the expression inside the parentheses: \[ 5\left(\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right) - 16 \] 3. **Distribute the 5 and simplify:** \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \cdot \frac{16}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} - \frac{80}{5} \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] 4. **Write the final completed square form:** \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Thus, the completed square form of the quadratic expression \(5x^2 - 8x - 16\) is: \[ \boxed{5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5}} \] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| deepseek-r1 | To complete the square for the quadratic \(5x^2 - 8x - 16\): 1. Factor out the coefficient of \(x^2\) from the first two terms: \[ 5(x^2 - \frac{8}{5}x) - 16 \] 2. To complete the square inside the parentheses, take half of the coefficient of \(x\), which is \(-\frac{4}{5}\), and square it to get \(\left(-\frac{4}{5}\right)^2 = \frac{16}{25}\). Add and subtract this value inside the parentheses: \[ 5\left(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}\right) - 16 \] 3. Simplify the expression by factoring the perfect square trinomial and adjusting the constants: \[ 5\left(\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right) - 16 \] 4. Distribute the 5 and combine the constants: \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \cdot \frac{16}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} - \frac{80}{5} \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Thus, the quadratic \(5x^2 - 8x - 16\) in completed square form is \(\boxed{5\left(x - \dfrac{4}{5}\right)^2 - \dfrac{96}{5}}\). |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| gpto1 | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), we proceed step by step: **Step 1:** Factor out the coefficient of \(x^2\) from the \(x^2\) and \(x\) terms: \[ 5x^2 - 8x - 16 = 5\left(x^2 - \frac{8}{5}x\right) - 16 \] **Step 2:** Complete the square inside the parentheses. Take half of the coefficient of \(x\) (which is \(-\frac{8}{5}\)): \[ \frac{1}{2} \times \left(-\frac{8}{5}\right) = -\frac{4}{5} \] Square this value: \[ \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \] Add and subtract this square inside the parentheses: \[ 5\left[\left(x^2 - \frac{8}{5}x + \frac{16}{25}\right) - \frac{16}{25}\right] - 16 \] **Step 3:** Simplify the expression. The quadratic inside the parentheses is now a perfect square: \[ 5\left[\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right] - 16 \] **Step 4:** Distribute the 5 and combine constants: \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \times \frac{16}{25} - 16 = 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} - 16 \] \[ \text{Since } -\frac{16}{5} - 16 = -\frac{16}{5} - \frac{80}{5} = -\frac{96}{5} \] **Final Answer:** \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Answer: \boxed{5\left( x\!-\!\dfrac{4}{5} \right)^{2}\ -\ \dfrac{96}{5}} |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| gpt-4o-2024-08-06 | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** The coefficient of \(x^2\) is 5. We factor 5 out of the terms involving \(x\): \[ 5x^2 - 8x = 5(x^2 - \frac{8}{5}x) \] 2. **Complete the square inside the parentheses:** To complete the square for the expression \(x^2 - \frac{8}{5}x\), we need to find a constant that makes it a perfect square trinomial. The formula for completing the square is: \[ x^2 + bx \rightarrow \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 \] Here, \(b = -\frac{8}{5}\). Calculate \(\frac{b}{2}\): \[ \frac{-\frac{8}{5}}{2} = -\frac{4}{5} \] Now, square \(-\frac{4}{5}\): \[ \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \] Add and subtract \(\frac{16}{25}\) inside the parentheses: \[ x^2 - \frac{8}{5}x = \left(x - \frac{4}{5}\right)^2 - \frac{16}{25} \] 3. **Substitute back into the expression:** Substitute the completed square back into the expression: \[ 5(x^2 - \frac{8}{5}x) = 5\left(\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right) \] Distribute the 5: \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \cdot \frac{16}{25} = 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} \] 4. **Combine with the constant term:** The original expression is \(5x^2 - 8x - 16\). We now combine \(-\frac{16}{5}\) with \(-16\): Convert \(-16\) to a fraction with a denominator of 5: \[ -16 = -\frac{80}{5} \] Combine the constants: \[ -\frac{16}{5} - \frac{80}{5} = -\frac{96}{5} \] 5. **Write the final expression:** The completed square form of the quadratic is: \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Therefore, the completed square form of the quadratic expression is: \[ \boxed{5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5}} \] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| DeepSeek-Coder-V2-Lite-Instruct | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 5x^2 - 8x - 16 = 5(x^2 - \frac{8}{5}x) - 16 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\) inside the parentheses, which is \(-\frac{8}{5}\). - Divide it by 2: \(\frac{-\frac{8}{5}}{2} = -\frac{4}{5}\). - Square this result: \(\left(-\frac{4}{5}\right)^2 = \frac{16}{25}\). 3. **Add and subtract this square inside the parentheses:** \[ 5\left(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}\right) - 16 \] \[ = 5\left(\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right) - 16 \] 4. **Distribute the 5 and simplify:** \[ = 5\left(x - \frac{4}{5}\right)^2 - 5 \cdot \frac{16}{25} - 16 \] \[ = 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - 16 \] \[ = 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - \frac{400}{25} \] \[ = 5\left(x - \frac{4}{5}\right)^2 - \frac{480}{25} \] \[ = 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Thus, the completed square form of the quadratic expression \(5x^2 - 8x - 16\) is: \[ \boxed{5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5}} \] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the quadratic equation The given quadratic equation is $5x^2 - 8x - 16$. ## Step 2: Divide the entire equation by the coefficient of $x^2$ to make the coefficient of $x^2$ equal to 1 Dividing the entire equation by 5 gives $x^2 - \frac{8}{5}x - \frac{16}{5}$. ## Step 3: Move the constant term to the right side of the equation Moving the constant term gives $x^2 - \frac{8}{5}x = \frac{16}{5}$. ## Step 4: Take half of the coefficient of $x$, square it, and add it to both sides of the equation Half of the coefficient of $x$ is $-\frac{4}{5}$. Squaring it gives $\frac{16}{25}$. Adding it to both sides gives $x^2 - \frac{8}{5}x + \frac{16}{25} = \frac{16}{5} + \frac{16}{25}$. ## Step 5: Simplify the right side of the equation Simplifying the right side gives $x^2 - \frac{8}{5}x + \frac{16}{25} = \frac{80}{25} + \frac{16}{25}$, which simplifies to $x^2 - \frac{8}{5}x + \frac{16}{25} = \frac{96}{25}$. ## Step 6: Factor the left side of the equation into a perfect square Factoring the left side gives $\left(x - \frac{4}{5}\right)^2 = \frac{96}{25}$. ## Step 7: Write the final answer in the required format The final answer is: $\boxed{\left(x - \frac{4}{5}\right)^2 = \frac{96}{25}}$ |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the coefficients of the quadratic equation The given quadratic equation is $5x^2 - 8x - 16$. The coefficients are $a = 5$, $b = -8$, and $c = -16$. ## 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 $5$, from the first two terms: $5x^2 - 8x = 5(x^2 - \frac{8}{5}x)$. ## Step 3: Find the value to complete the square To complete the square, we need to add and subtract $(\frac{b}{2})^2$ inside the parentheses. In this case, $\frac{b}{2} = \frac{-8}{2 \cdot 5} = -\frac{4}{5}$. So, $(\frac{b}{2})^2 = (-\frac{4}{5})^2 = \frac{16}{25}$. ## Step 4: Add and subtract the value to complete the square We add and subtract $\frac{16}{25}$ inside the parentheses: $5(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25})$. ## Step 5: Write the expression as a perfect square and simplify Now, we can write the expression as a perfect square: $5(x^2 - \frac{8}{5}x + \frac{16}{25}) - 5 \cdot \frac{16}{25} - 16$. This simplifies to $5(x - \frac{4}{5})^2 - \frac{80}{25} - 16$. ## Step 6: Simplify the constant terms Combine the constant terms: $-\frac{80}{25} - 16 = -\frac{80}{25} - \frac{400}{25} = -\frac{480}{25} = -\frac{96}{5}$. ## Step 7: Write the final expression in completed square form The final expression in completed square form is $5(x - \frac{4}{5})^2 - \frac{96}{5}$. The final answer is: $\boxed{5(x - \frac{4}{5})^2 - \frac{96}{5}}$ |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| Qwen2-72B-Instruct | To complete the square for the quadratic expression $5x^2 - 8x - 16$, we first factor out the coefficient of $x^2$ from the first two terms: \[5x^2 - 8x - 16 = 5(x^2 - \frac{8}{5}x) - 16\] Next, we find the value that completes the square for the expression inside the parentheses. This is done by taking half of the coefficient of $x$ (which is $\frac{8}{5}$), squaring it, and adding it inside the parentheses. Half of $\frac{8}{5}$ is $\frac{4}{5}$, and squaring this gives $\left(\frac{4}{5}\right)^2 = \frac{16}{25}$. We add and subtract this value inside the parentheses to maintain equality: \[= 5\left(x^2 - \frac{8}{5}x + \left(\frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2\right) - 16\] \[= 5\left[\left(x - \frac{4}{5}\right)^2 - \left(\frac{4}{5}\right)^2\right] - 16\] \[= 5\left(x - \frac{4}{5}\right)^2 - 5\left(\frac{16}{25}\right) - 16\] \[= 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - 16\] \[= 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - \frac{400}{25}\] \[= 5\left(x - \frac{4}{5}\right)^2 - \frac{480}{25}\] \[= 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5}\] Therefore, the completed square form of $5x^2 - 8x - 16$ is: \[\boxed{5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5}}\] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| baichuan4 | To complete the square for the quadratic $5x^2 - 8x - 16$, we first need to identify the coefficients of the quadratic terms and the linear term. The coefficient of the quadratic term is $5$, and the coefficient of the linear term is $-8$. Our goal is to rewrite the quadratic in the form $(ax + b)^2 + c$, where $a$, $b$, and $c$ are constants. To do this, we need to ensure that the $x^2$ and $x$ terms in the expanded form of $(ax + b)^2$ match the corresponding terms in the original quadratic. The expanded form of $(ax + b)^2$ is $a^2x^2 + 2abx + b^2$. We want the coefficient of $x^2$ in this expansion to match the coefficient of $x^2$ in the original quadratic, which is $5$. Therefore, we need $a^2 = 5$, so $a = \sqrt{5}$. Next, we want the coefficient of $x$ in the expansion to match the coefficient of $x$ in the original quadratic, which is $-8$. In the expansion, the coefficient of $x$ is $2ab$. We have $a = \sqrt{5}$, so we need $2ab = -8$. Solving for $b$, we get $b = \frac{-8}{2\sqrt{5}} = \frac{-4}{\sqrt{5}}$. Now we can write the quadratic in the form $(ax + b)^2 + c$. We have $a = \sqrt{5}$ and $b = \frac{-4}{\sqrt{5}}$, so the square part is $(\sqrt{5}x - \frac{4}{\sqrt{5}})^2$. To find $c$, we need to calculate the constant term in the expanded form of $(\sqrt{5}x - \frac{4}{\sqrt{5}})^2$ and compare it to the constant term in the original quadratic, which is $-16$. Expanding $(\sqrt{5}x - \frac{4}{\sqrt{5}})^2$, we get: $$ (\sqrt{5}x - \frac{4}{\sqrt{5}})^2 = (\sqrt{5}x)^2 - 2(\sqrt{5}x)(\frac{4}{\sqrt{5}}) + (\frac{4}{\sqrt{5}})^2 $$ $$ = 5x^2 - 8x + \frac{16}{5} $$ Comparing this to the original quadratic $5x^2 - 8x - 16$, we see that the constant term in the expanded square is $\frac{16}{5}$, while the constant term in the original quadratic is $-16$. Therefore, $c = -16 - \frac{16}{5} = -\frac{80}{5} - \frac{16}{5} = -\frac{96}{5}$. Putting it all together, we can write the original quadratic as: $$ 5x^2 - 8x - 16 = (\sqrt{5}x - \frac{4}{\sqrt{5}})^2 - \frac{96}{5} $$ However, we want to express $c$ in the form $k - \frac{16}{5}$ for some constant $k$. We have $c = -\frac{96}{5}$, so $k = -\frac{96}{5} + \frac{16}{5} = -\frac{80}{5} = -16$. Therefore, the final answer is: $$ 5x^2 - 8x - 16 = (\sqrt{5}x - \frac{4}{\sqrt{5}})^2 - 16 $$ We can check our answer by expanding the square and verifying that the resulting expression matches the original quadratic. |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| doubao | $5x^2 - 8x - 16$ $5(x^2 - \frac{8}{5}x) - 16$ $5(x^2 - \frac{8}{5}x + (\frac{8}{5}\div2)^2 - (\frac{8}{5}\div2)^2) - 16$ $5(x^2 - \frac{8}{5}x + (\frac{4}{5})^2 - (\frac{4}{5})^2) - 16$ $5((x - \frac{4}{5})^2 - \frac{16}{25}) - 16$ $5(x - \frac{4}{5})^2 - 5\times\frac{16}{25} - 16$ $5(x - \frac{4}{5})^2 - \frac{16}{5} - 16$ $5(x - \frac{4}{5})^2 - \frac{16}{5} - \frac{80}{5}$ $5(x - \frac{4}{5})^2 - \frac{96}{5}$ $\boxed{5(x - \frac{4}{5})^2 - \frac{96}{5}}$ |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| gpto1 | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), we proceed step by step: **Step 1:** Factor out the coefficient of \(x^2\) from the \(x^2\) and \(x\) terms: \[ 5x^2 - 8x - 16 = 5\left(x^2 - \frac{8}{5}x\right) - 16 \] **Step 2:** Complete the square inside the parentheses. Take half of the coefficient of \(x\) (which is \(-\frac{8}{5}\)): \[ \frac{1}{2} \times \left(-\frac{8}{5}\right) = -\frac{4}{5} \] Square this value: \[ \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \] Add and subtract this square inside the parentheses: \[ 5\left[\left(x^2 - \frac{8}{5}x + \frac{16}{25}\right) - \frac{16}{25}\right] - 16 \] **Step 3:** Simplify the expression. The quadratic inside the parentheses is now a perfect square: \[ 5\left[\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right] - 16 \] **Step 4:** Distribute the 5 and combine constants: \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \times \frac{16}{25} - 16 = 5\left(x - \frac{4}{5}\right)^2 - \frac{16}{5} - 16 \] \[ \text{Since } -\frac{16}{5} - 16 = -\frac{16}{5} - \frac{80}{5} = -\frac{96}{5} \] **Final Answer:** \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{96}{5} \] Answer: \boxed{5\left( x\!-\!\dfrac{4}{5} \right)^{2}\ -\ \dfrac{96}{5}} |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| kimi | To complete the square for the quadratic expression \(5x^2 - 8x - 16\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 5x^2 - 8x - 16 = 5(x^2 - \frac{8}{5}x) - 16 \] 2. **Identify the coefficient of \(x\) inside the parentheses, which is \(-\frac{8}{5}\). To complete the square, we need to add and subtract the square of half of this coefficient:** \[ \left(\frac{-\frac{8}{5}}{2}\right)^2 = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \] 3. **Add and subtract \(\frac{16}{25}\) inside the parentheses:** \[ 5\left(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}\right) - 16 \] 4. **Rewrite the expression inside the parentheses as a perfect square trinomial and a constant term:** \[ 5\left((x - \frac{4}{5})^2 - \frac{16}{25}\right) - 16 \] 5. **Distribute the 5 through the parentheses:** \[ 5(x - \frac{4}{5})^2 - 5 \cdot \frac{16}{25} - 16 \] 6. **Simplify the constant terms:** \[ 5(x - \frac{4}{5})^2 - \frac{80}{25} - 16 \] \[ 5(x - \frac{4}{5})^2 - \frac{80}{25} - \frac{400}{25} \] \[ 5(x - \frac{4}{5})^2 - \frac{480}{25} \] \[ 5(x - \frac{4}{5})^2 - \frac{96}{5} \] Thus, the completed square form of the quadratic expression \(5x^2 - 8x - 16\) is: \[ \boxed{5(x - \frac{4}{5})^2 - \frac{96}{5}} \] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| minimax | To complete the square for the quadratic equation $5x^2 - 8x - 16$, we need to form a perfect square trinomial from the first two terms. Here are the steps: 1. Factor out the coefficient of $x^2$ from the first two terms: $$ 5(x^2 - \frac{8}{5}x) - 16 $$ 2. Find the value that completes the square for the expression in the parentheses. This value is $(\frac{b}{2a})^2$, where $a$ is the coefficient of $x^2$ (which is 1 after factoring out the 5) and $b$ is the coefficient of $x$ (which is $-\frac{8}{5}$ in this case). So we calculate: $$ \left(\frac{-\frac{8}{5}}{2 \cdot 1}\right)^2 = \left(\frac{-8}{10}\right)^2 = \left(\frac{-4}{5}\right)^2 = \frac{16}{25} $$ 3. Add and subtract this value inside the parentheses to complete the square: $$ 5\left(x^2 - |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| wenxinyiyan | 首先，我们需要将二次项系数化为1，这样我们就可以更轻松地完成平方。我们可以通过将整个表达式除以5来实现这一点： $5x^2 - 8x - 16 = 5(x^2 - \frac{8}{5}x) - 16$ 接下来，我们需要找到一个数，使得$x^2 - \frac{8}{5}x$可以变成一个完全平方。这个数应该是线性项系数的一半的平方，即$(\frac{8}{10})^2 = \frac{16}{25}$。 然后，我们将这个数加到表达式中并减去相同的数，以保持等式的平衡： $5(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}) - 16$ 现在我们可以完成平方了： $5((x - \frac{4}{5})^2 - \frac{16}{25}) - 16$ 最后，我们可以将常数项合并，得到最终答案： $\boxed{5(x - \frac{4}{5})^2 - \frac{104}{5}}$ |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| zhipu | To complete the square for the quadratic $5x^2 - 8x - 16$, follow these steps: 1. **Factor out the leading coefficient** from the terms involving $x$: \[ 5x^2 - 8x - 16 = 5(x^2 - \frac{8}{5}x) - 16 \] 2. **Complete the square** inside the parentheses: - Take half of the coefficient of $x$, square it, and add and subtract it inside the parentheses: \[ \left(\frac{-\frac{8}{5}}{2}\right)^2 = \left(-\frac{4}{5}\right)^2 = \frac{16}{25} \] - Add and subtract this square inside the parentheses: \[ 5\left(x^2 - \frac{8}{5}x + \frac{16}{25} - \frac{16}{25}\right) - 16 \] - Simplify the expression inside the parentheses: \[ 5\left(\left(x - \frac{4}{5}\right)^2 - \frac{16}{25}\right) - 16 \] 3. **Distribute the 5** and combine the constants: \[ 5\left(x - \frac{4}{5}\right)^2 - 5 \cdot \frac{16}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - \frac{80}{25} - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - 3.2 - 16 \] \[ 5\left(x - \frac{4}{5}\right)^2 - 19.2 \] Thus, the completed square form of the quadratic $5x^2 - 8x - 16$ is: \[ \boxed{5\left(x - \frac{4}{5}\right)^2 - 19.2} \] |
5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |
| deepseek-v3 | Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-9b57063a-84e9-98f4-b6fd-f6d517167b64', 'request_id': '9b57063a-84e9-98f4-b6fd-f6d517167b64'} | 5 \left(x-\frac{4}{5}\right)^2-\frac{96}{5} |