omplete the square to identify the two values that solve the following quadratic equation: x^2+8x=9 .(1 point)

Answer 1

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

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

2. Rewrite the quadratic expression: To complete the square, we need to add and subtract the square of half the coefficient of \( x \) (which is 8). Half of 8 is 4, and squaring it gives us \( 4^2 = 16 \). We add and subtract 16 on the left side: \[ x^2 + 8x + 16 - 16 = 9 \] This simplifies to: \[ (x + 4)^2 - 16 = 9 \]

3. Move -16 to the other side of the equation: \[ (x + 4)^2 = 9 + 16 \] \[ (x + 4)^2 = 25 \]

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

5. Solve for \( x \):

   • For the positive case: \[ x + 4 = 5 \implies x = 1 \]
   • For the negative case: \[ x + 4 = -5 \implies x = -9 \]

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