Question

which equation is related to \\(\sqrt{x + 10} - 1 = x\\)?
options:
\\(x + 10 = x^2 + x + 1\\)
\\(x + 10 = x^2 - 1\\)
\\(x + 10 = x^2 + 2x + 1\\)
\\(x + 10 = x^2 + 1\\)

Explanation:

Step1: Isolate the square root

Start with the equation \(\sqrt{x + 10}-1=x\). Add 1 to both sides to isolate the square root: \(\sqrt{x + 10}=x + 1\).

Step2: Square both sides

Square both sides of the equation \(\sqrt{x + 10}=x + 1\) to eliminate the square root. Recall that \((a + b)^2=a^2+2ab + b^2\), so squaring the left side gives \(x + 10\) and squaring the right side gives \((x + 1)^2=x^2+2x + 1\). Thus, we get \(x + 10=x^2+2x + 1\).

Answer:

The equation related to \(\sqrt{x + 10}-1 = x\) is \(x + 10=x^2+2x + 1\) (the second option from the left, or the option with the equation \(x + 10=x^2+2x + 1\)).