Complete the square to solve the following quadratic equation: x2+2x−8=0 .(1 point) Responses x=−2, x=4 x equals negative 2 , x equals 4 x=−10, x=8 x equals negative 10 , x equals 8 x=2, x=−4 x equals 2 , x equals negative 4 x=10, x=−8

Question

GPT-4o mini · Answer

To solve the quadratic equation $ x^2 + 2x - 8 = 0 $ by completing the square, follow these steps: 1. Move the constant to the right side of the equation: $$ x^2 + 2x = 8 $$ 2. To complete the square, take half of the coefficient of $ x $ (which is $ 2 $), square it, and add it to both sides. Half of $ 2 $ is $ 1 $, and $ 1^2 = 1 $: $$ x^2 + 2x + 1 = 8 + 1 $$ $$ x^2 + 2x + 1 = 9 $$ 3. The left side can now be factored: $$ (x + 1)^2 = 9 $$ 4. Now, take the square root of both sides: $$ x + 1 = 3 \quad \text{or} \quad x + 1 = -3 $$ 5. Solve for $ x $: $$ x + 1 = 3 \quad \Rightarrow \quad x = 2 $$ $$ x + 1 = -3 \quad \Rightarrow \quad x = -4 $$ Thus, the solutions are: $$ x = 2 \quad \text{and} \quad x = -4 $$ So, the correct answer is: **x equals 2, x equals negative 4**.

Honeybeee · Answer

Complete the square to solve the following quadratic equation: x2−22x+1=15 .(1 point) Responses x=−11+14−−√, −11−14−−√ x equals negative 11 plus Start Root 14 End Root comma negative 11 minus Start Root 14 End Root x=−11+315−−√, −11−315−−√ x equals negative 11 plus 3 Start Root 15 End Root comma negative 11 minus 3 Start Root 15 End Root x=11+14−−√, 11−14−−√ x equals 11 plus Start Root 14 End Root comma 11 minus Start Root 14 End Root x=11+315−−√, 11−315−−√

GPT-4o mini · Answer

To solve the given quadratic equation $ x^2 - 22x + 1 = 15 $ by completing the square, we will follow these steps: 1. **Rearrange the equation**: Start by moving the constant term from the right side: $$ x^2 - 22x + 1 - 15 = 0 $$ Simplifying gives: $$ x^2 - 22x - 14 = 0 $$ 2. **Move the constant to the right side**: $$ x^2 - 22x = 14 $$ 3. **Complete the square**: Take half of the coefficient of $ x $ (which is $ -22 $), square it, and add it to both sides. Half of $ -22 $ is $ -11 $, and $ -11^2 = 121 $: $$ x^2 - 22x + 121 = 14 + 121 $$ $$ x^2 - 22x + 121 = 135 $$ 4. **Factor the left side**: $$ (x - 11)^2 = 135 $$ 5. **Take the square root of both sides**: $$ x - 11 = \sqrt{135} \quad \text{or} \quad x - 11 = -\sqrt{135} $$ 6. **Solve for $ x $**: $$ x = 11 + \sqrt{135} \quad \text{or} \quad x = 11 - \sqrt{135} $$ Now, since $ \sqrt{135} = \sqrt{9 \cdot 15} = 3\sqrt{15} $, the final solutions are: $$ x = 11 + 3\sqrt{15} \quad \text{and} \quad x = 11 - 3\sqrt{15} $$ Therefore, the correct response is: **x equals 11 plus 3 Start Root 15 End Root, 11 minus 3 Start Root 15 End Root**.