Question

radical equations and extraneous roots
which equation is equivalent to \\(\sqrt{x^2 + 81} = x + 10\\)?
\\(x^2 + 81 = x^2 + 100\\)
\\(x + 9 = x^2 + 20x + 100\\)
\\(x + 9 = x + 10\\)
\\(x^2 + 81 = x^2 + 20x + 100\\)

Explanation:

Step1: Square both sides of the equation

To eliminate the square root, we square both sides of the equation \(\sqrt{x^2 + 81}=x + 10\). Recall that \((\sqrt{a})^2=a\) for \(a\geq0\) and \((a + b)^2=a^2+2ab + b^2\). So, squaring the left - hand side gives \((\sqrt{x^2 + 81})^2=x^2 + 81\), and squaring the right - hand side gives \((x + 10)^2=x^2+2\times x\times10 + 10^2=x^2 + 20x+100\).

Step2: Write the equivalent equation

After squaring both sides, the equation \(\sqrt{x^2 + 81}=x + 10\) is equivalent to \(x^2 + 81=x^2 + 20x + 100\).

Answer:

\(x^{2}+81 = x^{2}+20x + 100\) (the fourth option)