Question

solve for x in the equation $x^2 - 10x + 25 = 35$. $x = 5pm 2sqrt{5}$ $x = 5pm sqrt{35}$ $x = 10pm 2sqrt{5}$ $x = 10pm sqrt{35}$

Explanation:

Step1: Factor the left - hand side

The left - hand side of the equation \(x^{2}-10x + 25\) is a perfect square trinomial. Recall that \((a - b)^{2}=a^{2}-2ab + b^{2}\). For \(x^{2}-10x + 25\), we have \(a=x\) and \(2ab = 10x\), so \(b = 5\) (since \(2\times x\times5=10x\)). Then \(x^{2}-10x + 25=(x - 5)^{2}\). The equation becomes \((x - 5)^{2}=35\).

Step2: Take the square root of both sides

Take the square root of both sides of the equation \((x - 5)^{2}=35\). We know that if \(y^{2}=k\) (\(k\geq0\)), then \(y=\pm\sqrt{k}\). So, taking the square root of both sides gives \(x - 5=\pm\sqrt{35}\).

Step3: Solve for x

Add 5 to both sides of the equation \(x - 5=\pm\sqrt{35}\) to isolate \(x\). We get \(x=5\pm\sqrt{35}\).

Answer:

\(x = 5\pm\sqrt{35}\) (the second option: \(x=5\pm\sqrt{35}\))