| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | To factor the quadratic expression \(-5x^2 - 100x\), we start by factoring out the common factor from both terms. 1. Identify the common factor: \[ -5x^2 - 100x \] Both terms have a common factor of \(-5x\). 2. Factor out \(-5x\): \[ -5x(x + 20) \] Thus, the factored form of the quadratic expression is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |
| deepseek-r1 | To factor the quadratic expression \(-5x^2 - 100x\), we start by identifying the greatest common factor (GCF) of the terms \(-5x^2\) and \(-100x\). The coefficients are \(-5\) and \(-100\). The GCF of 5 and 100 is 5. Since both coefficients are negative, we factor out \(-5x\) instead of \(5x\) to make the remaining polynomial have a positive leading coefficient. Factoring out \(-5x\) from each term: \[ -5x^2 - 100x = -5x \left( \frac{-5x^2}{-5x} + \frac{-100x}{-5x} \right) \] Simplifying each term inside the parentheses: \[ \frac{-5x^2}{-5x} = x \quad \text{and} \quad \frac{-100x}{-5x} = 20 \] Thus, the factored form is: \[ -5x(x + 20) \] Verifying by expanding \(-5x(x + 20)\): \[ -5x \cdot x + -5x \cdot 20 = -5x^2 - 100x \] which matches the original expression. The factored form of the quadratic is \(\boxed{-5x(x + 20)}\). |
5 (-x-20) x |
| gpto1 | \boxed{-5x\, (x + 20)} | 5 (-x-20) x |
| gpt-4o-2024-08-06 | To factor the quadratic expression \(-5x^2 - 100x\), we start by identifying the greatest common factor (GCF) of the terms. The terms are \(-5x^2\) and \(-100x\). The coefficients are \(-5\) and \(-100\), and the variable part is \(x^2\) and \(x\). 1. **Find the GCF of the coefficients:** - The GCF of \(-5\) and \(-100\) is \(5\). 2. **Find the GCF of the variable part:** - The variable part is \(x^2\) and \(x\). The GCF is \(x\). 3. **Combine the GCFs:** - The overall GCF of the expression is \(5x\). Now, factor out the GCF from each term: \[ -5x^2 - 100x = -5x(x) - 5x(20) \] This can be rewritten as: \[ -5x(x + 20) \] Thus, the factored form of the quadratic expression is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |
| DeepSeek-Coder-V2-Lite-Instruct | To factor the quadratic expression \(-5x^2 - 100x\), we can follow these steps: 1. **Identify the common factor**: Look for the greatest common factor (GCF) of the terms in the expression. The terms are \(-5x^2\) and \(-100x\). 2. **Factor out the GCF**: The GCF of \(-5x^2\) and \(-100x\) is \(-5x\). 3. **Rewrite the expression**: Factor \(-5x\) out of each term in the expression. \[ -5x^2 - 100x = -5x(x) - 5x(20) \] 4. **Combine the factored terms**: Combine the terms inside the parentheses. \[ -5x(x + 20) \] Thus, the factored form of the quadratic expression \(-5x^2 - 100x\) is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |
| Meta-Llama-3.1-405B-Instruct-Turbo | ## Step 1: Identify the common factor in the quadratic expression The given quadratic expression is $-5x^2 - 100x$. We can see that both terms have a common factor of $-5x$. ## Step 2: Factor out the common factor from the quadratic expression Factoring out $-5x$ from both terms gives us $-5x(x + 20)$. The final answer is: $\boxed{-5x(x + 20)}$ |
5 (-x-20) x |
| Meta-Llama-3.1-70B-Instruct-Turbo | ## Step 1: Identify the common factor in the quadratic expression The given quadratic expression is $-5x^2 - 100x$. We can see that both terms have a common factor of $-5x$. ## Step 2: Factor out the common factor We can factor out $-5x$ from both terms to simplify the expression. Factoring out $-5x$ from $-5x^2$ gives us $-5x \cdot x$, and factoring out $-5x$ from $-100x$ gives us $-5x \cdot 20$. ## Step 3: Write the factored form of the quadratic expression After factoring out the common factor $-5x$, the expression becomes $-5x(x + 20)$. The final answer is: $\boxed{-5x(x + 20)}$ |
5 (-x-20) x |
| Qwen2-72B-Instruct | To factor the given quadratic expression, we first identify the common factors of the terms. The expression is: \[ -5x^2 - 100x \] Both terms have a common factor of \(-5x\). We can factor this out: \[ -5x(x + 20) \] So, the factored form of the given quadratic expression is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |
| baichuan4 | To factor the quadratic expression $-5x^2 - 100x$, we first look for a common factor that can be extracted from both terms. In this case, the number $-5$ is common to both terms, as it divides $-5x^2$ and $-100x$. We begin by factoring out the common factor: $$-5x^2 - 100x = -5(x^2 + 20x)$$ Next, we observe that the expression inside the parentheses, $x^2 + 20x$, can be further factored by identifying the two terms as having a common factor of $x$. We extract this common factor: $$-5(x^2 + 20x) = -5x(x + 20)$$ Now, we have fully factored the original quadratic expression. The final answer is: $$\boxed{-5x(x + 20)}$$ This expression represents the factored form of the quadratic $-5x^2 - 100x$. |
5 (-x-20) x |
| doubao | First, we can factor out the greatest common factor of $-5x$ from both terms: $-5x^2 - 100x = -5x(x + 20)$. So the answer is $\boxed{-5x(x+20)}$. | 5 (-x-20) x |
| gpto1 | \boxed{-5x\, (x + 20)} | 5 (-x-20) x |
| kimi | First, we factor out the greatest common factor, which is $-5x$: \[ -5 x^2 - 100 x = -5x(x + 20) \] Thus, the factored form is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |
| minimax | To factor the quadratic $-5x^2 - 100x$, we look for two numbers that multiply to give the product of the coefficient of $x^2$ (which is -5) and the constant term (which is 0 in this case), and at the same time, these two numbers should add up to the coefficient of $x$ (which is -100). However, since the constant term is 0, we can immediately see that the two numbers we are looking for are both 0. This means that the quadratic can be factored as follows: $$-5x^2 - 100x = -5x(x + 20)$$ So the factored form of the quadratic is: $$\boxed{-5x(x + 20)}$$ |
5 (-x-20) x |
| wenxinyiyan | 首先，观察给定的二次多项式 $-5x^2 - 100x$，可以发现它们的公因式是 $-5x$。 提取公因式，有： $-5x^2 - 100x = -5x \times (x + 20).$这样，我们就得到了因式分解的形式。 所以，答案是： $\boxed{-5x(x + 20)}.$ |
5 (-x-20) x |
| zhipu | First, we identify the common factor in both terms of the quadratic $-5x^2 - 100x$. The common factor is $-5x$. Factoring out $-5x$ from each term, we get: \[ -5x^2 - 100x = -5x(x) - 5x(20) \] This simplifies to: \[ -5x(x + 20) \] Thus, the factored form of the quadratic $-5x^2 - 100x$ is $\\boxed{-5x(x + 20)}$. |
5 (-x-20) x |
| deepseek-v3 | To factor the quadratic expression \(-5x^2 - 100x\), follow these steps: 1. **Factor out the Greatest Common Factor (GCF):** The GCF of \(-5x^2\) and \(-100x\) is \(-5x\). \[ -5x^2 - 100x = -5x(x + 20) \] 2. **Simplify the Expression:** The expression is already in its simplest factored form. Therefore, the factored form of \(-5x^2 - 100x\) is: \[ \boxed{-5x(x + 20)} \] |
5 (-x-20) x |