| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | We have that \begin{align*} 3^{x^2+4x+4}&=9^{x+2}\\ \Rightarrow\qquad 3^{x^2+4x+4}&=3^{2x+4}\\ \Rightarrow\qquad x^2+4x+4&=2x+4\\ \Rightarrow\qquad x^2+2x&=0\\ \Rightarrow\qquad x(x+2)&=0\\ \Rightarrow\qquad x&=0,-2. \end{align*}Therefore, the sum of all possible values of $x$ is $0+(-2)=\boxed{-2}$. The best answer is C. |
C |
| deepseekr1 | To solve the equation \(3^{x^2 + 4x + 4} = 9^{x + 2}\), we first express both sides with the same base. Since \(9 = 3^2\), the right side becomes \((3^2)^{x + 2} = 3^{2(x + 2)} = 3^{2x + 4}\). Equating the exponents: \[ x^2 + 4x + 4 = 2x + 4 \] Simplify by subtracting \(2x + 4\) from both sides: \[ x^2 + 2x = 0 \] Factor: \[ x(x + 2) = 0 \] The solutions are \(x = 0\) and \(x = -2\). Summing these values gives \(0 + (-2) = -2\). Verification: - For \(x = 0\), both sides equal \(3^4 = 81\). - For \(x = -2\), both sides equal \(3^0 = 1\). Thus, the sum of all possible \(x\) is **-2**. **Answer:** C |
C |
| gpto1 | C | C |
| GPT-4o | To solve the equation \( 3^{x^2 + 4x + 4} = 9^{x + 2} \), we first note that the base 9 can be rewritten in terms of base 3, since \( 9 = 3^2 \). Therefore, the equation becomes: \[ 3^{x^2 + 4x + 4} = (3^2)^{x + 2} \] \[ 3^{x^2 + 4x + 4} = 3^{2(x + 2)} \] Since the bases are the same, we can set the exponents equal to each other: \[ x^2 + 4x + 4 = 2(x + 2) \] Next, we simplify the right-hand side of the equation: \[ x^2 + 4x + 4 = 2x + 4 \] Subtract \( 2x + 4 \) from both sides to set the equation to zero: \[ x^2 + 4x + 4 - 2x - 4 = 0 \] \[ x^2 + 2x = 0 \] Factor out the common term \( x \): \[ x(x + 2) = 0 \] This gives us two possible solutions for \( x \): \[ x = 0 \quad \text{or} \quad x = -2 \] The sum of all possible values of \( x \) is: \[ 0 + (-2) = -2 \] Thus, the answer is: C. -2 |
C |
| llama3.1-70B-Instruct | ## Step 1: Rewrite the equation with the same base We can rewrite the equation $3^{x^2+4x+4}=9^{x+2}$ using the fact that $9=3^2$. This gives us $3^{x^2+4x+4}=(3^2)^{x+2}=3^{2x+4}$. ## Step 2: Equate the exponents Since the bases are the same, we can equate the exponents: $x^2+4x+4=2x+4$. ## Step 3: Solve the quadratic equation We can simplify the equation by subtracting $2x+4$ from both sides, giving us $x^2+2x=0$. Factoring out $x$, we get $x(x+2)=0$. ## Step 4: Find the solutions This gives us two possible solutions: $x=0$ and $x=-2$. ## Step 5: Calculate the sum of the solutions The sum of the solutions is $0+(-2)=-2$. The final answer is: $\boxed{C}$ |
C |
| llama3.1-405B-Instruct | We have that \begin{align*} 3^{x^2+4x+4}&=9^{x+2}\\ \Rightarrow\qquad 3^{x^2+4x+4}&=3^{2x+4}\\ \Rightarrow\qquad x^2+4x+4&=2x+4\\ \Rightarrow\qquad x^2+2x&=0\\ \Rightarrow\qquad x(x+2)&=0\\ \Rightarrow\qquad x&=0,-2. \end{align*}Therefore, the sum of all possible values of $x$ is $0+(-2)=\boxed{-2}$. The best answer is C. |
C |
| DeepSeek-Coder-V2 | C | C |
| Qwen2-72B-Instruct | C. -2 Explanation: We can rewrite $9^{x+2}$ as $(3^2)^{x+2}=3^{2(x+2)}$. So the equation becomes $3^{x^2+4x+4}=3^{2(x+2)}$. Since the bases are equal, the exponents must be equal. So we have $x^2+4x+4=2(x+2)$. Simplifying this equation gives $x^2+2x=0$. Factoring out an x gives $x(x+2)=0$. So the solutions are $x=0$ and $x=-2$. The sum of these solutions is $0+(-2)=-2$. |
C |
| baichuan4 | B | C |
| doubao | We can rewrite $9$ as $3^2$. So $9^{x+2}=(3^2)^{x+2}=3^{2(x+2)}=3^{2x+4}$. Then the equation becomes $3^{x^2+4x+4}=3^{2x+4}$. So $x^2+4x+4=2x+4$. Rearranging gives $x^2+2x=0$, which factors as $x(x+2)=0$. So $x=0$ or $x=-2$. The sum of the possible values is $0+(-2)=-2$. So the answer is C. | C |
| minimax | To solve the equation $3^{x^2+4x+4}=9^{x+2}$, we can start by expressing the right side of the equation in terms of base 3, since 9 is a power of 3. We know that $9 = 3^2$, so we can rewrite the equation as: $3^{x^2+4x+4} = (3^2)^{x+2}$ Now, apply the power rule $(a^b)^c = a^{bc}$ to the right side: $3^{x^2+4x+4} = 3^{2(x+2)}$ This simplifies to: $3^{x^2+4x+4} = 3^{2x+4}$ Since the bases are the same, we can set the exponents equal to each other: $x^2 + 4x + 4 = 2x + 4$ Now, we can solve for x by subtracting 2x from both sides and subtracting 4 from both sides: $x^2 + 4x + 4 - 2x - 4 = 2x + 4 - 2x - 4$ $x^2 + 2x = 0$ Factor out an x: $x(x + 2) = 0$ This gives us two solutions: $x = 0$ or $x + 2 = 0 \Rightarrow x = -2$ The sum of all possible values of x is: $0 + (-2) = -2$ So the answer is: C. -2 |
C |
| kimi | To solve the equation \(3^{x^2+4x+4}=9^{x+2}\), we can start by expressing both sides of the equation with the same base. Since \(9\) can be written as \(3^2\), we can rewrite the equation as: \[3^{x^2+4x+4} = (3^2)^{x+2}\] Now, we can apply the exponent rule \((a^m)^n = a^{mn}\) to the right side: \[3^{x^2+4x+4} = 3^{2(x+2)}\] Next, we can simplify the exponent on the right side: \[3^{x^2+4x+4} = 3^{2x+4}\] Since the bases are the same, we can set the exponents equal to each other: \[x^2 + 4x + 4 = 2x + 4\] Now, we can solve for \(x\) by subtracting \(2x + 4\) from both sides: \[x^2 + 4x + 4 - 2x - 4 = 0\] \[x^2 + 2x = 0\] Factor out an \(x\) from the left side: \[x(x + 2) = 0\] This gives us two possible solutions for \(x\): \[x = 0 \quad \text{or} \quad x = -2\] The sum of all possible values of \(x\) is \(0 + (-2) = -2\). Answer: C. -2 |
C |
| wenxinyiyan | 首先，将原方程 $3^{x^2+4x+4}=9^{x+2}$ 化为同一底数，即 $3^{x^2+4x+4}=3^{2(x+2)}$。 然后，由于底数相同，我们可以直接比较指数，得到 $x^2+4x+4=2(x+2)$。 化简后得到 $x^2+2x=0$，解这个二次方程得到 $x=0$ 或 $x=-2$。 最后，求这两个解的和，即 $0+(-2)=-2$。 故答案为：C. $-2$。 |
C |
| zhipu | To solve for the sum of all possible values of \( x \) in the equation \( 3^{x^2 + 4x + 4} = 9^{x + 2} \), we first express both sides of the equation with the same base. Recall that \( 9 \) can be written as \( 3^2 \). Therefore, we can rewrite the equation as: \[ 3^{x^2 + 4x + 4} = (3^2)^{x + 2} \] Using the property of exponents \((a^m)^n = a^{mn}\), we can simplify the right-hand side: \[ (3^2)^{x + 2} = 3^{2(x + 2)} = 3^{2x + 4} \] Now, the equation becomes: \[ 3^{x^2 + 4x + 4} = 3^{2x + 4} \] Since the bases are the same, we can set the exponents equal to each other: \[ x^2 + 4x + 4 = 2x + 4 \] Subtract \( 2x + 4 \) from both sides to set the equation to zero: \[ x^2 + 4x + 4 - 2x - 4 = 0 \] \[ x^2 + 2x = 0 \] Factor the quadratic equation: \[ x(x + 2) = 0 \] This gives us two solutions: \[ x = 0 \] \[ x = -2 \] The sum of these possible values of \( x \) is: \[ 0 + (-2) = -2 \] Therefore, the correct answer is: C. -2 |
C |
| gpto1 | C | C |
| deepseekv3 | To solve the equation \( 3^{x^2 + 4x + 4} = 9^{x + 2} \), follow these steps: 1. **Express both sides with the same base:** - Since \( 9 = 3^2 \), we can rewrite the right side: \[ 9^{x + 2} = (3^2)^{x + 2} = 3^{2(x + 2)} \] - Now the equation becomes: \[ 3^{x^2 + 4x + 4} = 3^{2(x + 2)} \] 2. **Set the exponents equal:** - Since the bases are the same, the exponents must be equal: \[ x^2 + 4x + 4 = 2(x + 2) \] 3. **Simplify the equation:** - Expand the right side: \[ x^2 + 4x + 4 = 2x + 4 \] - Move all terms to the left side: \[ x^2 + 2x = 0 \] - Factor the equation: \[ x(x + 2) = 0 \] 4. **Find the solutions:** - Set each factor equal to zero: \[ x = 0 \quad \text{or} \quad x = -2 \] 5. **Calculate the sum of all possible values of \( x \):** \[ 0 + (-2) = -2 \] However, let's double-check the simplification step to ensure accuracy. **Revisiting Step 2:** \[ x^2 + 4x + 4 = 2(x + 2) \] Expanding the right side: \[ x^2 + 4x + 4 = 2x + 4 \] Subtract \( 2x + 4 \) from both sides: \[ x^2 + 2x = 0 \] Factor: \[ x(x + 2) = 0 \] Solutions: \[ x = 0 \quad \text{or} \quad x = -2 \] Sum of solutions: \[ 0 + (-2) = -2 \] Therefore, the correct answer is: \boxed{C} |
C |