Question

you try! 1. $x^2 - 16x = -15$ 2. $x^2 - 18x = -17$ example: solve each equation by completing the square. 1 $x^2 + -2x = 3$ 2. $x^2 + 12x = -8$

Explanation:

Step1: Rewrite the equation (1)

For the equation \(x^{2}-16x = - 15\), 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 \(- 16\), half of it is \(\frac{-16}{2}=-8\), and squaring it gives \((-8)^{2} = 64\).
Add 64 to both sides of the equation:
\(x^{2}-16x + 64=-15 + 64\)

Step2: Factor the left side

The left side is a perfect square trinomial, so it factors as \((x - 8)^{2}\). The right side simplifies to \(49\):
\((x - 8)^{2}=49\)

Step3: Take square roots

Take the square root of both sides:
\(x-8=\pm\sqrt{49}=\pm7\)

Step4: Solve for \(x\)

Case 1: \(x - 8=7\), then \(x=7 + 8=15\)
Case 2: \(x - 8=-7\), then \(x=-7 + 8 = 1\)

Step1: Rewrite the equation (2)

For the equation \(x^{2}-18x=-17\), the coefficient of \(x\) is \(-18\), half of it is \(\frac{-18}{2}=-9\), and squaring it gives \((-9)^{2}=81\).
Add 81 to both sides of the equation:
\(x^{2}-18x + 81=-17 + 81\)

Step2: Factor the left side

The left side factors as \((x - 9)^{2}\), and the right side simplifies to \(64\):
\((x - 9)^{2}=64\)

Step3: Take square roots

Take the square root of both sides:
\(x - 9=\pm\sqrt{64}=\pm8\)

Step4: Solve for \(x\)

Case 1: \(x - 9 = 8\), then \(x=8 + 9=17\)
Case 2: \(x - 9=-8\), then \(x=-8 + 9 = 1\)

Answer:

For \(x^{2}-16x=-15\), \(x = 1\) or \(x = 15\); for \(x^{2}-18x=-17\), \(x = 1\) or \(x = 17\)