Question

1. ((x + 2)^2 + 3 = 51)

Explanation:

Step1: Subtract 3 from both sides

To isolate the squared term, we subtract 3 from both sides of the equation \((x + 2)^2 + 3 = 51\). This gives us \((x + 2)^2 = 51 - 3\), so \((x + 2)^2 = 48\).

Step2: Take square root of both sides

Taking the square root of both sides, we get \(x + 2=\pm\sqrt{48}\). Simplify \(\sqrt{48}\) as \(\sqrt{16\times3} = 4\sqrt{3}\), so \(x + 2=\pm4\sqrt{3}\).

Step3: Solve for x

Subtract 2 from both sides to solve for \(x\). We have two solutions: \(x=- 2 + 4\sqrt{3}\) and \(x=-2 - 4\sqrt{3}\). If we want decimal approximations, \(4\sqrt{3}\approx4\times1.732 = 6.928\), so \(x\approx - 2+6.928 = 4.928\) or \(x\approx - 2 - 6.928=-8.928\).

Answer:

\(x=-2 + 4\sqrt{3}\) or \(x=-2 - 4\sqrt{3}\) (or approximately \(x\approx4.93\) or \(x\approx - 8.93\))