| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To complete the square for the quadratic expression \(13x^2 - 13x - 6\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 13x^2 - 13x - 6 = 13(x^2 - x) - 6 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-1\), divide it by 2, and square it: \[ \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \] - Add and subtract this square inside the parentheses: \[ 13(x^2 - x + \frac{1}{4} - \frac{1}{4}) - 6 \] - This can be rewritten as: \[ 13\left((x^2 - x + \frac{1}{4}) - \frac{1}{4}\right) - 6 \] - Simplify the expression inside the parentheses: \[ 13\left((x - \frac{1}{2})^2 - \frac{1}{4}\right) - 6 \] 3. **Distribute the 13 and simplify:** \[ 13(x - \frac{1}{2})^2 - 13 \cdot \frac{1}{4} - 6 \] \[ 13(x - \frac{1}{2})^2 - \frac{13}{4} - 6 \] 4. **Combine the constants:** - Convert \(-6\) to a fraction with the same denominator as \(-\frac{13}{4}\): \[ -6 = -\frac{24}{4} \] - Add the fractions: \[ -\frac{13}{4} - \frac{24}{4} = -\frac{37}{4} \] 5. **Write the final expression:** \[ 13(x - \frac{1}{2})^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic expression \(13x^2 - 13x - 6\) is: \[ \boxed{13(x - \frac{1}{2})^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| deepseek-r1 | To complete the square for the quadratic \(13x^2 - 13x - 6\), we start by factoring out the coefficient of \(x^2\) from the first two terms: \[ 13(x^2 - x) - 6 \] Next, we need to complete the square inside the parentheses. The coefficient of \(x\) is \(-1\). We take half of this coefficient, which is \(-\frac{1}{2}\), and square it to get \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). We add and subtract this value inside the parentheses: \[ 13\left[(x^2 - x + \frac{1}{4} - \frac{1}{4})\right] - 6 \] This can be rewritten as: \[ 13\left[(x^2 - x + \frac{1}{4}) - \frac{1}{4}\right] - 6 \] Simplifying further, we have: \[ 13\left(x - \frac{1}{2}\right)^2 - 13 \cdot \frac{1}{4} - 6 \] Calculating the constants: \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - 6 \] Converting 6 to a fraction with a denominator of 4: \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - \frac{24}{4} \] Combining the constants: \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4} \] Thus, the quadratic \(13x^2 - 13x - 6\) in completed square form is: \[ \boxed{13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| gpto1 | To complete the square for the quadratic \(13x^2 - 13x - 6\), follow these steps: **Step 1: Factor out the coefficient of \(x^2\) from the first two terms.** \(13x^2 - 13x - 6 = 13(x^2 - x) - 6\) **Step 2: Complete the square inside the parentheses.** For the expression \(x^2 - x\): - Take half of the coefficient of \(x\): \(-1 \div 2 = -\dfrac{1}{2}\) - Square it: \(\left(-\dfrac{1}{2}\right)^2 = \dfrac{1}{4}\) Add and subtract \(\dfrac{1}{4}\) inside the parentheses to complete the square: \(13\left(x^2 - x + \dfrac{1}{4} - \dfrac{1}{4}\right) - 6\) Simplify inside the parentheses: \(13\left((x - \dfrac{1}{2})^2 - \dfrac{1}{4}\right) - 6\) **Step 3: Distribute the 13 and combine constants.** Multiply 13 with both terms inside the parentheses: \(13(x - \dfrac{1}{2})^2 - 13 \times \dfrac{1}{4} - 6\) Simplify: \(13(x - \dfrac{1}{2})^2 - \dfrac{13}{4} - 6\) Combine the constants: \(-\dfrac{13}{4} - 6 = -\dfrac{13}{4} - \dfrac{24}{4} = -\dfrac{37}{4}\) **Final Answer:** \(\boxed{13\left(x - \dfrac{1}{2}\right)^{2} - \dfrac{37}{4}}\) |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| gpt-4o-2024-08-06 | To complete the square for the quadratic \(13x^2 - 13x - 6\), we first focus on the quadratic and linear terms: \(13x^2 - 13x\). 1. Factor out the coefficient of \(x^2\) from the first two terms: \[ 13(x^2 - x) \] 2. To complete the square inside the parentheses, take the coefficient of \(x\), which is \(-1\), divide it by 2, and square it: \[ \left(-\frac{1}{2}\right)^2 = \frac{1}{4} \] 3. Add and subtract this square inside the parentheses: \[ 13(x^2 - x + \frac{1}{4} - \frac{1}{4}) \] 4. Rewrite the expression by completing the square: \[ 13\left((x - \frac{1}{2})^2 - \frac{1}{4}\right) \] 5. Distribute the 13: \[ 13(x - \frac{1}{2})^2 - 13 \times \frac{1}{4} = 13(x - \frac{1}{2})^2 - \frac{13}{4} \] 6. Don't forget the constant term \(-6\) from the original quadratic: \[ 13(x - \frac{1}{2})^2 - \frac{13}{4} - 6 \] 7. Convert \(-6\) to a fraction with a denominator of 4: \[ -6 = -\frac{24}{4} \] 8. Combine the constant terms: \[ -\frac{13}{4} - \frac{24}{4} = -\frac{37}{4} \] 9. The completed square form of the quadratic is: \[ 13(x - \frac{1}{2})^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic \(13x^2 - 13x - 6\) is: \[ \boxed{13(x - \frac{1}{2})^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| DeepSeek-Coder-V2-Lite-Instruct | To complete the square for the quadratic expression \(13 x^2 - 13 x - 6\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 13 x^2 - 13 x - 6 = 13 (x^2 - x) - 6 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-1\), divide it by 2, and square it: \[ \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \] - Add and subtract this square inside the parentheses: \[ 13 (x^2 - x + \frac{1}{4} - \frac{1}{4}) - 6 \] - This can be rewritten as: \[ 13 \left(x^2 - x + \frac{1}{4}\right) - 13 \cdot \frac{1}{4} - 6 \] 3. **Simplify the expression:** - The term inside the parentheses is a perfect square: \[ x^2 - x + \frac{1}{4} = \left(x - \frac{1}{2}\right)^2 \] - Substitute this back into the expression: \[ 13 \left(x - \frac{1}{2}\right)^2 - 13 \cdot \frac{1}{4} - 6 \] - Simplify the constant terms: \[ 13 \left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - 6 \] - Combine the constants: \[ - \frac{13}{4} - 6 = - \frac{13}{4} - \frac{24}{4} = - \frac{37}{4} \] 4. **Write the final expression:** \[ 13 \left(x - \frac{1}{2}\right)^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic expression \(13 x^2 - 13 x - 6\) is: \[ \boxed{13 \left(x - \frac{1}{2}\right)^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the coefficients of the quadratic equation The given quadratic equation is $13x^2 - 13x - 6$. The coefficients are $a = 13$, $b = -13$, and $c = -6$. ## Step 2: Calculate the value to complete the square To complete the square, we need to add and subtract $(\frac{b}{2})^2$ inside the parentheses. First, calculate this value: $(\frac{-13}{2})^2 = \frac{169}{4}$. ## Step 3: Rewrite the quadratic equation to complete the square Start rewriting the equation by factoring out the coefficient of $x^2$, which is 13: $13(x^2 - x) - 6$. Now, add and subtract $\frac{169}{4}$ inside the parentheses: $13(x^2 - x + \frac{169}{4} - \frac{169}{4}) - 6$. ## Step 4: Write the completed square and simplify Now, write the expression inside the parentheses as a perfect square and simplify the rest: $13((x - \frac{1}{2})^2 - \frac{169}{4}) - 6$. Simplify further: $13(x - \frac{1}{2})^2 - \frac{169 \cdot 13}{4} - 6$. Calculate the constants: $-\frac{169 \cdot 13}{4} - 6 = -\frac{2197}{4} - \frac{24}{4} = -\frac{2221}{4}$. ## Step 5: Write the final form of the completed square Combine the terms: $13(x - \frac{1}{2})^2 - \frac{2221}{4}$. The final answer is: $\boxed{13(x - \frac{1}{2})^2 - \frac{2221}{4}}$ |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the coefficients of the quadratic equation The given quadratic equation is $13x^2 - 13x - 6$. The coefficients are $a = 13$, $b = -13$, and $c = -6$. ## Step 2: Move the constant term to the right side of the equation To complete the square, we need to isolate the $x^2$ and $x$ terms on one side. So, we move the constant term to the right side: $13x^2 - 13x = 6$. ## Step 3: Factor out the coefficient of $x^2$ from the left side We factor out $13$ from the left side to make the coefficient of $x^2$ equal to $1$: $13(x^2 - x) = 6$. ## Step 4: Calculate the value to complete the square To complete the square, we need to add $(\frac{b}{2})^2$ to both sides. Since $b = -13$, we calculate $(\frac{-13}{2})^2 = (\frac{13}{2})^2 = \frac{169}{4}$. ## Step 5: Add the calculated value to both sides We add $\frac{169}{4}$ to both sides of the equation: $13(x^2 - x + \frac{169}{4 \cdot 13}) = 6 + \frac{169}{4}$. ## Step 6: Simplify the left side to a perfect square The left side simplifies to $13(x - \frac{1}{2})^2$. ## Step 7: Simplify the right side The right side simplifies to $6 + \frac{169}{4} = \frac{24}{4} + \frac{169}{4} = \frac{193}{4}$. ## Step 8: Write the completed square form The completed square form is $13(x - \frac{1}{2})^2 = \frac{193}{4}$. ## Step 9: Divide by the coefficient of $x^2$ to isolate the squared term Divide by $13$ to isolate the squared term: $(x - \frac{1}{2})^2 = \frac{193}{4 \cdot 13}$. ## Step 10: Simplify the right side The right side simplifies to $(x - \frac{1}{2})^2 = \frac{193}{52}$. ## Step 11: Further simplify the right side The right side further simplifies to $(x - \frac{1}{2})^2 = \frac{193}{52} = \frac{193 \div 1}{52 \div 1} = \frac{193}{52}$. ## Step 12: Reduce the fraction on the right side Since $193$ and $52$ have no common factors, the fraction cannot be reduced further. The final answer is: $\boxed{(x - \frac{1}{2})^2 = \frac{193}{52}}$ |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| Qwen2-72B-Instruct | To complete the square for the quadratic equation $13x^2 - 13x - 6$, we first factor out the coefficient of $x^2$ from the first two terms: \[13(x^2 - x) - 6\] Next, we need to 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 $-1$), squaring it, and adding it inside the parentheses. Half of $-1$ is $-\frac{1}{2}$, and squaring this gives $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$. However, since we're adding $\frac{1}{4}$ inside the parentheses, which is being multiplied by $13$ outside, we must also subtract $13 \times \frac{1}{4}$ outside the parentheses to keep the equation balanced. Thus, our equation becomes: \[13\left(x^2 - x + \frac{1}{4}\right) - 6 - 13\left(\frac{1}{4}\right)\] Simplifying further: \[13\left(x - \frac{1}{2}\right)^2 - 6 - \frac{13}{4}\] \[13\left(x - \frac{1}{2}\right)^2 - \frac{24}{4} - \frac{13}{4}\] \[13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4}\] Therefore, the completed square form of the given quadratic equation is: \[\boxed{13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4}}\] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| baichuan4 | To complete the square for the quadratic $13x^2 - 13x - 6$, we first need to divide the entire expression by the coefficient of the $x^2$ term, which is $13$. This gives us: $$ x^2 - x - \frac{6}{13} $$ Next, we want to focus on the $x$ term. To complete the square, we need to add and subtract the square of half of the coefficient of the $x$ term. The coefficient of $x$ is $-1$, so half of it is $-\frac{1}{2}$. The square of this is $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$. Now, we add $\frac{1}{4}$ to both sides of the equation: $$ x^2 - x + \frac{1}{4} - \frac{6}{13} + \frac{1}{4} $$ The last two terms can be combined: $$ x^2 - x + \frac{1}{4} - \frac{6}{13} + \frac{1}{4} = x^2 - x + \frac{1}{4} + \frac{5}{52} $$ $$ x^2 - x + \frac{1}{4} + \frac{5}{52} = x^2 - x + \frac{13}{52} + \frac{5}{52} $$ $$ x^2 - x + \frac{18}{52} = x^2 - x + \frac{9}{26} $$ Now, we can factor the left side of the equation as a perfect square: $$ (x - \frac{1}{2})^2 = x^2 - x + \frac{9}{26} $$ Finally, we need to remember that we divided the original expression by $13$, so we need to multiply both sides of the equation by $13$ to get back to the original expression: $$ 13(x - \frac{1}{2})^2 = 13(x^2 - x + \frac{9}{26}) $$ $$ 13(x - \frac{1}{2})^2 = 13x^2 - 13x + \frac{9}{2} $$ $$ 13(x - \frac{1}{2})^2 = 13x^2 - 13x + \frac{18}{2} $$ $$ 13(x - \frac{1}{2})^2 = 13x^2 - 13x + 9 $$ So, the original expression $13x^2 - 13x - 6$ can be rewritten as: $$ 13(x - \frac{1}{2})^2 - 9 $$ This is not quite the form we want, because we need to have the constant term inside the box. To do this, we can rewrite the equation as: $$ 13[(x - \frac{1}{2})^2 - \frac{9}{13}] $$ And now we can put the constant term inside the box: $$ 13[(x - \frac{1}{2})^2 - \frac{9}{13}] = 13[(x - \frac{1}{2})^2 - \frac{3}{13}] $$ $$ 13[(x - \frac{1}{2})^2 - \frac{3}{13}] = 13[(x - \frac{1}{2})^2 - \frac{6}{13}] $$ So, the completed square form of the original expression is: $$ \boxed{13(x - \frac{1}{2})^2 - \frac{6}{13}} $$ However, this is not quite right because we need to have the entire expression inside the box. To do this, we can distribute the $13$ inside the box: $$ \boxed{13(x - \frac{1}{2})^2 - 6} $$ And this is the completed square form of the original expression. |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| doubao | $13x^2 - 13x - 6$ $=13(x^2 - x) - 6$ $=13(x^2 - x + \frac{1}{4} - \frac{1}{4}) - 6$ $=13[(x - \frac{1}{2})^2 - \frac{1}{4}] - 6$ $=13(x - \frac{1}{2})^2 - 13 \times \frac{1}{4} - 6$ $=13(x - \frac{1}{2})^2 - \frac{13}{4} - \frac{24}{4}$ $=13(x - \frac{1}{2})^2 - \frac{37}{4}$ $\boxed{13(x - \frac{1}{2})^2 - \frac{37}{4}}$ |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| gpto1 | To complete the square for the quadratic \(13x^2 - 13x - 6\), follow these steps: **Step 1: Factor out the coefficient of \(x^2\) from the first two terms.** \(13x^2 - 13x - 6 = 13(x^2 - x) - 6\) **Step 2: Complete the square inside the parentheses.** For the expression \(x^2 - x\): - Take half of the coefficient of \(x\): \(-1 \div 2 = -\dfrac{1}{2}\) - Square it: \(\left(-\dfrac{1}{2}\right)^2 = \dfrac{1}{4}\) Add and subtract \(\dfrac{1}{4}\) inside the parentheses to complete the square: \(13\left(x^2 - x + \dfrac{1}{4} - \dfrac{1}{4}\right) - 6\) Simplify inside the parentheses: \(13\left((x - \dfrac{1}{2})^2 - \dfrac{1}{4}\right) - 6\) **Step 3: Distribute the 13 and combine constants.** Multiply 13 with both terms inside the parentheses: \(13(x - \dfrac{1}{2})^2 - 13 \times \dfrac{1}{4} - 6\) Simplify: \(13(x - \dfrac{1}{2})^2 - \dfrac{13}{4} - 6\) Combine the constants: \(-\dfrac{13}{4} - 6 = -\dfrac{13}{4} - \dfrac{24}{4} = -\dfrac{37}{4}\) **Final Answer:** \(\boxed{13\left(x - \dfrac{1}{2}\right)^{2} - \dfrac{37}{4}}\) |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| kimi | To complete the square for the quadratic expression \(13x^2 - 13x - 6\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 13x^2 - 13x - 6 = 13(x^2 - x) - 6 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-1\). - Divide it by 2: \(\frac{-1}{2} = -\frac{1}{2}\). - Square the result: \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). 3. **Add and subtract this square inside the parentheses:** \[ 13(x^2 - x + \frac{1}{4} - \frac{1}{4}) - 6 \] 4. **Rewrite the expression by grouping the perfect square trinomial and the remaining terms:** \[ 13\left(x^2 - x + \frac{1}{4}\right) - 13 \cdot \frac{1}{4} - 6 \] 5. **Simplify the expression inside the parentheses:** \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - 6 \] 6. **Combine the constants outside the parentheses:** \[ -\frac{13}{4} - 6 = -\frac{13}{4} - \frac{24}{4} = -\frac{37}{4} \] 7. **Write the final expression:** \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic expression \(13x^2 - 13x - 6\) is: \[ \boxed{13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| minimax | To complete the square for the quadratic equation $13x^2 - 13x - 6$, we need to express it in the form $(ax + b)^2 + c$. First, we factor out the coefficient of $x^2$ from the first two terms: $$13(x^2 - x) - 6$$ Next, we need to find a value to complete the square inside the parentheses. To do this, we take half of the coefficient of $x$ and square it. The coefficient of $x$ is $-1$, so half of $-1$ is $-\frac{1}{2}$, and squaring this gives us $\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$. We add and subtract this value inside the parentheses to complete the square: $$13\left(x^2 - x + \frac{1}{4}\right) - 13\left(\frac{1}{4}\right) - 6$$ Now we can rewrite the expression inside the parentheses as a perfect square: $$13\left(x - \frac{1}{2}\right)^2 - 13\left(\ |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| 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。给定的二次方程是 $13x^2 - 13x - 6$，我们可以将其除以13得到： $x^2 - x - \frac{6}{13}$ 接下来，我们需要找到一个数，使得 $x^2 - x$ 可以变成一个完全平方。这个数应该是二次项系数的一半，即 $-1/2$，然后我们需要计算这个数的平方，即 $(-1/2)^2 = 1/4$。 现在，我们可以将 $1/4$ 加到 $x^2 - x$ 上，然后再减去 $1/4$，以保持方程的平衡。这样我们得到： $x^2 - x + \frac{1}{4} - \frac{1}{4} - \frac{6}{13}$ 现在我们可以将前三项组合成一个完全平方项，得到： $(x - \frac{1}{2})^2 - \frac{1}{4} - \frac{6}{13}$ 最后，我们可以将常数项合并，得到最终答案： $\boxed{(x - \frac{1}{2})^2 - \frac{37}{52}}$ |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| zhipu | To complete the square for the quadratic $13x^2 - 13x - 6$, we follow these steps: 1. **Factor out the leading coefficient** from the terms involving $x^2$ and $x$: \[ 13x^2 - 13x - 6 = 13(x^2 - x) - 6 \] 2. **Complete the square** inside the parentheses: - Take half of the coefficient of $x$, square it, and add and subtract this square inside the parentheses: \[ x^2 - x = x^2 - x + \left(\frac{1}{2}\right)^2 - \left(\frac{1}{2}\right)^2 = \left(x - \frac{1}{2}\right)^2 - \frac{1}{4} \] 3. **Substitute back** into the original expression: \[ 13(x^2 - x) - 6 = 13\left(\left(x - \frac{1}{2}\right)^2 - \frac{1}{4}\right) - 6 \] 4. **Distribute the 13** and simplify: \[ 13\left(x - \frac{1}{2}\right)^2 - 13 \cdot \frac{1}{4} - 6 = 13\left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - 6 \] Convert $-6$ to a fraction with denominator 4: \[ -6 = -\frac{24}{4} \] So, the expression becomes: \[ 13\left(x - \frac{1}{2}\right)^2 - \frac{13}{4} - \frac{24}{4} = 13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic $13x^2 - 13x - 6$ is: \[ \boxed{13\left(x - \frac{1}{2}\right)^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |
| deepseek-v3 | To complete the square for the quadratic expression \(13x^2 - 13x - 6\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 13x^2 - 13x - 6 = 13(x^2 - x) - 6 \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\), which is \(-1\), divide it by 2, and square it: \[ \left(\frac{-1}{2}\right)^2 = \frac{1}{4} \] - Add and subtract this square inside the parentheses: \[ 13(x^2 - x + \frac{1}{4} - \frac{1}{4}) - 6 \] - This can be rewritten as: \[ 13\left((x^2 - x + \frac{1}{4}) - \frac{1}{4}\right) - 6 \] - Simplify the expression inside the parentheses: \[ 13\left((x - \frac{1}{2})^2 - \frac{1}{4}\right) - 6 \] 3. **Distribute the 13 and simplify:** \[ 13(x - \frac{1}{2})^2 - 13 \cdot \frac{1}{4} - 6 \] \[ 13(x - \frac{1}{2})^2 - \frac{13}{4} - 6 \] 4. **Combine the constants:** - Convert \(-6\) to a fraction with the same denominator as \(-\frac{13}{4}\): \[ -6 = -\frac{24}{4} \] - Add the fractions: \[ -\frac{13}{4} - \frac{24}{4} = -\frac{37}{4} \] 5. **Write the final expression:** \[ 13(x - \frac{1}{2})^2 - \frac{37}{4} \] Thus, the completed square form of the quadratic expression \(13x^2 - 13x - 6\) is: \[ \boxed{13(x - \frac{1}{2})^2 - \frac{37}{4}} \] |
13 \left(x-\frac{1}{2}\right)^2-\frac{37}{4} |