Question

1. ((x - 6)^2 = 40)

Explanation:

Step1: Take square root of both sides

To solve for \(x\), we first take the square root of both sides of the equation \((x - 6)^2=40\). Remember that when we take the square root of a number, we get both a positive and a negative root. So we have:
\(x - 6=\pm\sqrt{40}\)

Step2: Simplify \(\sqrt{40}\)

We can simplify \(\sqrt{40}\) by factoring 40. Since \(40 = 4\times10\), and \(\sqrt{4\times10}=\sqrt{4}\times\sqrt{10}=2\sqrt{10}\). So the equation becomes:
\(x - 6=\pm2\sqrt{10}\)

Step3: Solve for \(x\)

Now, we add 6 to both sides of the equation to isolate \(x\). This gives us two solutions:
For the positive root: \(x = 6 + 2\sqrt{10}\)
For the negative root: \(x = 6 - 2\sqrt{10}\)

Answer:

\(x = 6 + 2\sqrt{10}\) or \(x = 6 - 2\sqrt{10}\) (or approximately \(x\approx6 + 6.3246=12.3246\) and \(x\approx6 - 6.3246=-0.3246\) if decimal approximations are preferred)