Question

solve each equation by completing the square.
$25 = 26x - x^2$

Explanation:

Step1: Rearrange the equation

First, we rewrite the given equation \(25 = 26x - x^{2}\) in standard quadratic form \(ax^{2}+bx + c = 0\). We add \(x^{2}\) to both sides and subtract \(26x\) from both sides to get:
\(x^{2}-26x + 25=0\)
Then, we rewrite it as \(x^{2}-26x=- 25\)

Step2: Complete the square

For a quadratic equation of the form \(x^{2}+bx = c\), to complete the square, we take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is \(- 26\), half of it is \(\frac{-26}{2}=-13\), and squaring it gives \((-13)^{2} = 169\). We add \(169\) to both sides of the equation \(x^{2}-26x=-25\):
\(x^{2}-26x + 169=-25 + 169\)
The left - hand side is a perfect square trinomial, which factors as \((x - 13)^{2}\), and the right - hand side simplifies to \(144\). So we have:
\((x - 13)^{2}=144\)

Step3: Solve for x

We take the square root of both sides. Remember that \(\sqrt{144}=\pm12\), so:
\(x - 13=\pm12\)
Case 1: When \(x - 13 = 12\), we add \(13\) to both sides: \(x=12 + 13=25\)
Case 2: When \(x - 13=-12\), we add \(13\) to both sides: \(x=-12 + 13 = 1\)

Answer:

After completing the square, the equation is \((x - 13)^{2}=144\) and the solution is \(x = 1\); \(x = 25\)