Question

1. x² - 10x + 25 = 17

Explanation:

Step1: Recognize perfect square trinomial

The left - hand side \(x^{2}-10x + 25\) is a perfect square trinomial. We know that \((a - b)^{2}=a^{2}-2ab + b^{2}\). For \(x^{2}-10x + 25\), if \(a=x\) and \(2ab = 10x\), then \(b = 5\) (since \(2\times x\times5=10x\)) and \(b^{2}=25\). So \(x^{2}-10x + 25=(x - 5)^{2}\). The equation becomes \((x - 5)^{2}=17\).

Step2: Take square roots

Take the square root of both sides. Remember that if \(y^{2}=k\) (\(k\geq0\)), then \(y=\pm\sqrt{k}\). So for \((x - 5)^{2}=17\), we have \(x - 5=\pm\sqrt{17}\).

Step3: Solve for x

Add 5 to both sides of the equation \(x - 5=\pm\sqrt{17}\) to isolate \(x\). We get \(x=5\pm\sqrt{17}\).

Answer:

\(x = 5+\sqrt{17}\) or \(x = 5-\sqrt{17}\)