Question

1. $x^2 - 18x + 81 = 49$

Explanation:

Step1: Recognize the perfect square trinomial

The left side \(x^2 - 18x + 81\) is a perfect square trinomial, which can be written as \((x - 9)^2\) because \((a - b)^2=a^2 - 2ab + b^2\), here \(a = x\), \(2ab=18x\) so \(b = 9\) and \(b^2 = 81\). So the equation becomes \((x - 9)^2=49\).

Step2: Take square roots on both sides

Taking square roots of both sides, we get \(x - 9=\pm\sqrt{49}\). Since \(\sqrt{49} = 7\), this simplifies to \(x - 9=\pm7\).

Step3: Solve for x in two cases

Case 1: When \(x - 9 = 7\), add 9 to both sides: \(x=7 + 9=16\).
Case 2: When \(x - 9=-7\), add 9 to both sides: \(x=-7 + 9 = 2\).

Answer:

\(x = 16\) or \(x = 2\)