Question

(j) \\(\sqrt{x^2 - 10x + 25} = 5\\)

Explanation:

Step1: Simplify the radicand

Notice that \(x^2 - 10x + 25\) is a perfect square trinomial, which factors as \((x - 5)^2\). So the equation becomes \(\sqrt{(x - 5)^2}=5\).

Step2: Simplify the square root

The square root of a square is the absolute value, so \(\vert x - 5\vert = 5\).

Step3: Solve the absolute value equation

This gives two cases:

• Case 1: \(x - 5 = 5\)

Solving for \(x\), we add 5 to both sides: \(x = 5 + 5 = 10\).

• Case 2: \(x - 5 = -5\)

Solving for \(x\), we add 5 to both sides: \(x = -5 + 5 = 0\).

Step4: Check the solutions

• For \(x = 10\): \(\sqrt{10^2 - 10\times10 + 25}=\sqrt{100 - 100 + 25}=\sqrt{25}=5\), which is correct.
• For \(x = 0\): \(\sqrt{0^2 - 10\times0 + 25}=\sqrt{0 - 0 + 25}=\sqrt{25}=5\), which is also correct.

Answer:

\(x = 0\) or \(x = 10\)