Question

completing the square
solve by completing the square. type the answers from smallest to largest.
$x^2 - 10x + 9 = 0$
$x = \square\square$ or $x = \square$
type the smaller solution here! type the larger solution here!

Explanation:

Step1: Rearrange the equation

We start with the equation \(x^{2}-10x + 9=0\). First, we move the constant term to the right side of the equation: \(x^{2}-10x=- 9\).

Step2: Complete the square

To complete the square for the quadratic expression \(x^{2}-10x\), we take half of the coefficient of \(x\), which is \(\frac{- 10}{2}=-5\), and then square it: \((-5)^{2} = 25\). We add this value to both sides of the equation:
\(x^{2}-10x + 25=-9 + 25\)
The left side can now be written as a perfect square: \((x - 5)^{2}=16\)

Step3: Solve for \(x\)

Take the square root of both sides: \(x - 5=\pm\sqrt{16}=\pm4\)
Now, solve for \(x\) in two cases:

• Case 1: \(x - 5 = 4\), then \(x=4 + 5=9\)
• Case 2: \(x - 5=-4\), then \(x=-4 + 5 = 1\)

Answer:

\(x = 1\) or \(x = 9\)