| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To complete the square for the quadratic expression \(2x^2 + 10x\), follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\) inside the parentheses, which is 5, divide it by 2, and square it: \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] - Add and subtract this square inside the parentheses: \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) \] - This can be rewritten as: \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] 3. **Distribute the 2 to both terms inside the parentheses:** \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) = 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} \] \[ = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] Thus, the completed square form of the quadratic expression \(2x^2 + 10x\) is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| deepseek-r1 | To complete the square for the quadratic expression \(2x^2 + 10x\): 1. Factor out the coefficient of \(x^2\) from the first two terms: \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. Inside the parentheses, take the coefficient of \(x\), which is 5, divide it by 2, and square the result: \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] 3. Add and subtract this square inside the parentheses. Since we are adding \(\frac{25}{4}\) inside the parentheses which is multiplied by 2, we need to subtract \(2 \times \frac{25}{4}\) to keep the expression balanced: \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) = 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] 4. Distribute the 2 and simplify: \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \times \frac{25}{4} = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] Thus, the quadratic \(2x^2 + 10x\) completed the square is \(\boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}}\). |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| gpto1 | To complete the square for the quadratic \(2x^2 + 10x\), let's carefully perform each step. **Step 1: Factor out the coefficient of \(x^2\) if it's not 1.** Since the coefficient is 2, we factor it out: \[ 2x^2 + 10x = 2(x^2 + 5x) \] **Step 2: Complete the square inside the parentheses.** To complete the square for \(x^2 + 5x\), we use the formula: \[ \text{Complete the square by adding and subtracting }\left( \frac{b}{2} \right)^2 \] Here, \(b = 5\), so: \[ \left( \frac{5}{2} \right)^2 = \frac{25}{4} \] So, we rewrite the expression: \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) \] Simplify the expression inside the brackets: \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] **Step 3: Expand and simplify the constants.** Distribute the 2: \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] **Final Answer:** \[ \boxed{2\left(x + \dfrac{5}{2}\right)^2 - \dfrac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| gpt-4o-2024-08-06 | To complete the square for the quadratic expression \(2x^2 + 10x\), we first factor out the coefficient of \(x^2\) from the terms involving \(x\). Here, the coefficient is 2. Thus, we have: \[ 2x^2 + 10x = 2(x^2 + 5x) \] Next, we complete the square for the expression inside the parentheses, \(x^2 + 5x\). To do this, we take the coefficient of \(x\), which is 5, divide it by 2, and then square the result: \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] We add and subtract this square inside the parentheses: \[ x^2 + 5x = \left(x^2 + 5x + \frac{25}{4}\right) - \frac{25}{4} \] This can be rewritten as: \[ x^2 + 5x = \left(x + \frac{5}{2}\right)^2 - \frac{25}{4} \] Substituting back into the expression with the factor of 2, we have: \[ 2(x^2 + 5x) = 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] Distribute the 2: \[ = 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} \] \[ = 2\left(x + \frac{5}{2}\right)^2 - \frac{50}{4} \] \[ = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] Thus, the expression \(2x^2 + 10x\) completed to a square is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| DeepSeek-Coder-V2-Lite-Instruct | To complete the square for the quadratic expression \(2x^2 + 10x\), we follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. **Complete the square inside the parentheses:** - Take the coefficient of \(x\) (which is 5), divide it by 2, and square it: \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] - Add and subtract this square inside the parentheses: \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) \] - This can be rewritten as: \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] 3. **Distribute the 2 and simplify:** \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] Thus, the completed square form of the quadratic expression \(2x^2 + 10x\) is: \[ 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] So, the final answer is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the coefficient of x^2 The coefficient of $x^2$ in the given quadratic is 2. ## Step 2: Factor out the coefficient of x^2 To complete the square, we first factor out the coefficient of $x^2$ from the quadratic and linear terms. So, we factor out 2: $2(x^2 + 5x)$. ## Step 3: Calculate the number to complete the square To complete the square for $x^2 + 5x$, we take half of the coefficient of $x$, which is $\frac{5}{2}$, and then square it: $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$. ## Step 4: Add and subtract the calculated number inside the parenthesis Adding and subtracting $\frac{25}{4}$ inside the parenthesis to complete the square, we get: $2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right)$. ## Step 5: Rewrite the expression with the completed square Simplifying, we have: $2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right)$. ## Step 6: Distribute the 2 Distributing the 2, we get: $2\left(x + \frac{5}{2}\right)^2 - 2\left(\frac{25}{4}\right)$. ## Step 7: Simplify the expression Simplifying further, we get: $2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}$. The final answer is: $\boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}}$ |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the coefficient of the x^2 term and the coefficient of the x term. The coefficient of the x^2 term is 2 and the coefficient of the x term is 10. ## Step 2: Factor out the coefficient of the x^2 term from the quadratic and the linear term. Factoring out 2 from the quadratic and linear terms gives: $2(x^2+5x)$. ## Step 3: Find the number that needs to be added to the quadratic to complete the square. To complete the square, we need to add $(\frac{5}{2})^2$ inside the parenthesis, because $\frac{5}{2}$ is half of the coefficient of the x term. ## Step 4: Add and subtract $(\frac{5}{2})^2$ inside the parenthesis to complete the square. Adding and subtracting $(\frac{5}{2})^2$ gives: $2(x^2+5x+(\frac{5}{2})^2-(\frac{5}{2})^2)$. ## Step 5: Write the completed square and simplify. Simplifying gives: $2(x^2+5x+(\frac{5}{2})^2)-2(\frac{5}{2})^2 = 2(x+\frac{5}{2})^2-\frac{25}{2}$. ## Step 6: Write the final answer in the required format. The final answer is: $\boxed{2(x+\frac{5}{2})^2-\frac{25}{2}}$ |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| Qwen2-72B-Instruct | To complete the square for the quadratic expression $2x^2 + 10x$, we need to add and subtract the square of half the coefficient of $x$ inside the expression. First, factor out the leading coefficient from the first two terms: \[2x^2 + 10x = 2(x^2 + 5x)\] Next, take half of the coefficient of $x$ (which is now 5) and square it: \[\left(\frac{5}{2}\right)^2 = \frac{25}{4}\] Add and subtract this value inside the parentheses, keeping in mind that since there's a factor of 2 outside the parentheses, you're actually adding and subtracting $2 \cdot \frac{25}{4} = \frac{50}{4} = 12.5$ to the entire expression: \[2(x^2 + 5x + \frac{25}{4} - \frac{25}{4}) = 2((x^2 + 5x + \frac{25}{4}) - \frac{25}{4})\] Rewrite the expression as a perfect square trinomial plus a constant term: \[2((x + \frac{5}{2})^2 - \frac{25}{4})\] Distribute the 2 back into the expression: \[2(x + \frac{5}{2})^2 - 2 \cdot \frac{25}{4}\] Simplify the constant term: \[2(x + \frac{5}{2})^2 - \frac{50}{4}\] \[2(x + \frac{5}{2})^2 - \frac{25}{2}\] So, the completed square form of $2x^2 + 10x$ is: \[\boxed{2(x + \frac{5}{2})^2 - \frac{25}{2}}\] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| baichuan4 | To complete the square for the quadratic $2x^2+10x$, we need to follow these steps: 1. Separate the quadratic into its coefficients: We have $a=2$, $b=10$, and $c=0$ (since there is no constant term). 2. Divide the $x$-term coefficient by $2a$ and square it: This gives us the value that we need to add to both sides of the equation to complete the square. $$\left(\frac{b}{2a}\right)^2 =left(\fracleft(\frac{10}{2 \cdot 2}\right)^2 = \left(\frac{10}{4}\right)^2 = \left(\frac{5}{2}\right)^2 = \frac{25}{4}$$ 3. Add the calculated value to both sides of the equation: $$2x^2 + 10x + \frac{25}{4} = \frac{25}{4}$$ 4. Factor the left side of the equation to complete the square: $$2\left(x^2 + 5x + \frac{25}{16}\right) = \frac{25}{4}$$ Note that we can factor out the $2$ because it's a common factor on the left side. 5. Simplify the equation if necessary: $$2\left(x + \frac{5}{4}\right)^2 = \frac{25}{4}$$ Now, we have completed the square for the quadratic $2x^2+10x$. The final answer is: $$\boxed{2\left(x + \frac{5}{4}\right)^2 = \frac{25}{4}}$$ This equation shows the quadratic in its completed square form. |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| doubao | $2x^2+10x = 2(x^2+5x) = 2(x^2+5x+\frac{25}{4}-\frac{25}{4}) = 2[(x+\frac{5}{2})^2-\frac{25}{4}] = 2(x+\frac{5}{2})^2-2\times\frac{25}{4} = 2(x+\frac{5}{2})^2 - \frac{25}{2}$. The answer is $\boxed{2(x+\frac{5}{2})^2 - \frac{25}{2}}$. | 2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| gpto1 | To complete the square for the quadratic \(2x^2 + 10x\), let's carefully perform each step. **Step 1: Factor out the coefficient of \(x^2\) if it's not 1.** Since the coefficient is 2, we factor it out: \[ 2x^2 + 10x = 2(x^2 + 5x) \] **Step 2: Complete the square inside the parentheses.** To complete the square for \(x^2 + 5x\), we use the formula: \[ \text{Complete the square by adding and subtracting }\left( \frac{b}{2} \right)^2 \] Here, \(b = 5\), so: \[ \left( \frac{5}{2} \right)^2 = \frac{25}{4} \] So, we rewrite the expression: \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) \] Simplify the expression inside the brackets: \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] **Step 3: Expand and simplify the constants.** Distribute the 2: \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] **Final Answer:** \[ \boxed{2\left(x + \dfrac{5}{2}\right)^2 - \dfrac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| kimi | To complete the square for the quadratic expression \(2x^2 + 10x\), we will follow these steps: 1. **Factor out the coefficient of \(x^2\) from the first two terms:** \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. **Identify the coefficient of \(x\) inside the parentheses, which is 5. To complete the square, we need to add and subtract the square of half of this coefficient:** \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] 3. **Add and subtract \(\frac{25}{4}\) inside the parentheses:** \[ 2(x^2 + 5x + \frac{25}{4} - \frac{25}{4}) \] 4. **Group the perfect square trinomial and the constant term separately:** \[ 2\left((x^2 + 5x + \frac{25}{4}) - \frac{25}{4}\right) \] 5. **Rewrite the perfect square trinomial as a squared binomial:** \[ x^2 + 5x + \frac{25}{4} = \left(x + \frac{5}{2}\right)^2 \] 6. **Substitute this back into the expression:** \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] 7. **Distribute the 2 through the parentheses:** \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} \] 8. **Simplify the constant term:** \[ 2 \cdot \frac{25}{4} = \frac{50}{4} = \frac{25}{2} \] 9. **Combine all parts to get the final expression:** \[ 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] Thus, the completed square form of the quadratic \(2x^2 + 10x\) is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| minimax | To complete the square for the quadratic $2x^2 + 10x$, we need to form a perfect square trinomial from the given quadratic expression. Here are the steps: 1. Factor out the coefficient of $x^2$ from the first two terms: $$2(x^2 + 5x)$$ 2. To complete the square, we need to add and subtract the square of half the coefficient of $x$ inside the parentheses. The coefficient of $x$ inside the parentheses is 5, so half of that is $\frac{5}{2}$, and squaring that gives us $\left(\frac{5}{2}\right)^2 = \frac{25}{4}$. 3. Add and subtract $\frac{25}{4}$ inside the parentheses: $$2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right)$$ 4. Now, rewrite the expression as a perfect square trinomial and a constant term: $$2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right)$$ 5. Distribute the |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| wenxinyiyan | 首先，我们提取公因数$2$，得到： $2x^2 + 10x = 2(x^2 + 5x)$ 为了完成平方，我们需要加上和减去$5^2 = 25$（因为二次项系数已经化为$1$，所以直接取一次项系数的一半，即$5$，然后平方）： $2(x^2 + 5x) = 2(x^2 + 5x + 25 - 25)$ $= 2((x + 5)^2 - 25)$ $= 2(x + 5)^2 - 50$ 所以，经过补全平方后，我们得到： $\boxed{2(x + 5)^2 - 50}$ |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| zhipu | To complete the square for the quadratic $2x^2 + 10x$, follow these steps: 1. **Factor out the leading coefficient** of the $x^2$ term, which is 2: \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. **Complete the square** inside the parentheses: - Take half of the coefficient of the $x$ term (which is 5), square it, and add and subtract this square inside the parentheses: \[ x^2 + 5x = x^2 + 5x + \left(\frac{5}{2}\right)^2 - \left(\frac{5}{2}\right)^2 \] - This simplifies to: \[ x^2 + 5x + \frac{25}{4} - \frac{25}{4} \] - Rewrite the expression as a perfect square trinomial minus the extra term: \[ (x + \frac{5}{2})^2 - \frac{25}{4} \] 3. **Substitute back** into the original expression: \[ 2(x^2 + 5x) = 2\left((x + \frac{5}{2})^2 - \frac{25}{4}\right) \] 4. **Distribute the 2**: \[ 2\left((x + \frac{5}{2})^2 - \frac{25}{4}\right) = 2(x + \frac{5}{2})^2 - 2 \cdot \frac{25}{4} \] - Simplify the constant term: \[ 2(x + \frac{5}{2})^2 - \frac{50}{4} = 2(x + \frac{5}{2})^2 - \frac{25}{2} \] So, the completed square form of the quadratic $2x^2 + 10x$ is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |
| deepseek-v3 | To complete the square for the quadratic expression \( 2x^2 + 10x \), follow these steps: 1. **Factor out the coefficient of \( x^2 \) from the first two terms:** \[ 2x^2 + 10x = 2(x^2 + 5x) \] 2. **Complete the square inside the parentheses:** - **Take half of the coefficient of \( x \) and square it:** \[ \left(\frac{5}{2}\right)^2 = \frac{25}{4} \] - **Add and subtract this value inside the parentheses:** \[ 2\left(x^2 + 5x + \frac{25}{4} - \frac{25}{4}\right) \] - **Rewrite the perfect square trinomial:** \[ 2\left(\left(x + \frac{5}{2}\right)^2 - \frac{25}{4}\right) \] 3. **Distribute the 2 and simplify:** \[ 2\left(x + \frac{5}{2}\right)^2 - 2 \cdot \frac{25}{4} = 2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2} \] So, the expression \( 2x^2 + 10x \) completed in square form is: \[ \boxed{2\left(x + \frac{5}{2}\right)^2 - \frac{25}{2}} \] |
2 \left(x+\frac{5}{2}\right)^2-\frac{25}{2} |