Solved on Dec 06, 2023

Find the solution to the quadratic equation x2=8x+9x^{2} = 8x + 9 by completing the square.

STEP 1

Assumptions
1. We are given the quadratic equation x2=8x+9x^{2} = 8x + 9.
2. We need to complete the square to solve the equation.
3. Completing the square involves creating a perfect square trinomial on one side of the equation.

STEP 2

First, we rewrite the equation by moving the linear term to the left side.
x28x=9x^{2} - 8x = 9

STEP 3

To complete the square, we need to add a specific value to both sides of the equation to form a perfect square trinomial on the left side. The value to add is the square of half the coefficient of xx.

STEP 4

Calculate half the coefficient of xx.
12×8=4\frac{1}{2} \times -8 = -4

STEP 5

Square the result from the previous step to find the value to add to both sides of the equation.
(4)2=16(-4)^2 = 16

STEP 6

Add 16 to each side of the equation to complete the square.
x28x+16=9+16x^{2} - 8x + 16 = 9 + 16

STEP 7

Now the left side of the equation is a perfect square trinomial, and the right side is the sum of 9 and 16.
x28x+16=25x^{2} - 8x + 16 = 25
The next step to complete the square is to add 16 to each side.

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