To complete the square for the equation \( x^2 + 8x = 9 \), follow these steps:
-
Move the constant term to the right side of the equation: \[ x^2 + 8x - 9 = 0 \]
-
Rewrite the equation focusing on the left side: \[ x^2 + 8x = 9 \]
-
To complete the square, take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides: \[ \left(\frac{8}{2}\right)^2 = 4^2 = 16 \]
-
Add 16 to both sides of the equation: \[ x^2 + 8x + 16 = 9 + 16 \] \[ x^2 + 8x + 16 = 25 \]
-
Now, the left side can be factored as a perfect square: \[ (x + 4)^2 = 25 \]
-
Take the square root of both sides: \[ x + 4 = \pm 5 \]
-
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 quadratic equation \( x^2 + 8x = 9 \) are \( x = 1 \) and \( x = -9 \).