To complete the square for the quadratic equation \( x^2 + 8x = 9 \), follow these steps:
-
Move the constant to the other side of the equation: \[ x^2 + 8x - 9 = 0 \]
-
Rewrite \( x^2 + 8x \) in a way that allows us to complete the square. First, take half of the coefficient of \( x \) (which is 8), square it, and add it inside the equation:
- Half of 8 is 4, and squaring it gives us \( 4^2 = 16 \).
We can rewrite the equation by adding and subtracting 16 on the left side: \[ x^2 + 8x + 16 - 16 = 9 \]
This simplifies to: \[ (x + 4)^2 - 16 = 9 \]
-
Move the constant (-16) to the right side of the equation: \[ (x + 4)^2 = 9 + 16 \] \[ (x + 4)^2 = 25 \]
-
Take the square root of both sides: \[ x + 4 = \pm 5 \]
-
Solve for \( x \):
- For the positive case: \[ x + 4 = 5 \implies x = 5 - 4 = 1 \]
- For the negative case: \[ x + 4 = -5 \implies x = -5 - 4 = -9 \]
Thus, the two values that solve the equation are: \[ x = 1 \quad \text{and} \quad x = -9 \]
Finally, to fill in the blank: x=____ (you can write either 1 or -9 depending on the requirements for the answer)