Question

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

Explanation:

Step1: Isolate the square root

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

Step2: Square both sides

Square both sides of the equation to eliminate the square root. Recall that $(a + b)^2 = a^2 + 2ab + b^2$. Here, $a = x$ and $b = 1$, so $(x + 1)^2 = x^2 + 2x + 1$.
Squaring both sides: $(\sqrt{x + 10})^2 = (x + 1)^2$
Simplify both sides: $x + 10 = x^2 + 2x + 1$

Answer:

$x + 10 = x^2 + 2x + 1$ (the second option)