Complete the square to identify the two values that solve the following quadratic equation: x^2+8x=9

Question

.(1 point)
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Answer 1

To complete the square for the equation \( x^2 + 8x = 9 \), follow these steps:

1. Move the constant to the other side of the equation: \[ x^2 + 8x - 9 = 0 \]

2. To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2, and square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]

3. Add and subtract this square (16) on the left side: \[ x^2 + 8x + 16 - 16 = 9 \] \[ x^2 + 8x + 16 = 25 \]

4. Now, rewrite the left side as a square: \[ (x + 4)^2 = 25 \]

5. Take the square root of both sides: \[ x + 4 = \pm 5 \]

6. Solve for \( x \): \[ x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1 \] \[ x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9 \]

So the two solutions for the equation \( x^2 + 8x = 9 \) are: \[ x = 1 \quad \text{and} \quad x = -9 \]

Answer 2

Incorrect

Make sure to combine all constants using the inverse of addition or subtraction before determining the square root.

Answer 3

Let's solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square correctly step by step.

1. Move the constant to the left side of the equation: \[ x^2 + 8x - 9 = 0 \]

2. Rearranging gives: \[ x^2 + 8x = 9 \]

3. To complete the square, take the coefficient of \( x \) (which is 8), divide it by 2, and square it: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]

4. Add and subtract this value (16) on the left side: \[ x^2 + 8x + 16 - 16 = 9 \] or \[ x^2 + 8x + 16 = 25 \]

5. Now, rewrite the left side as a perfect square: \[ (x + 4)^2 = 25 \]

6. Now take the square root of both sides: \[ x + 4 = \pm 5 \]

7. Solve for \( x \):

   • For \( x + 4 = 5 \): \[ x = 5 - 4 = 1 \]
   • For \( x + 4 = -5 \): \[ x = -5 - 4 = -9 \]

Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are: \[ x = 1 \quad \text{and} \quad x = -9 \]

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