Question

hs : advanced algebra: concepts and connections - block (27.0831030)
adical equations and extraneous roots
which equation is related to \\(\sqrt{x + 10} - 1 = x\\)?
\\(x + 10 = x^2 - 1\\)
\\(x + 10 = x^2 + 1\\)
\\(x + 10 = x^2 + 2x + 1\\)
\\(x + 10 = x^2 + x + 1\\)

Explanation:

Step1: Isolate the radical term

Start with the original 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 $\sqrt{x + 10}=x + 1$. Recall that $(\sqrt{a})^2=a$ and $(a + b)^2=a^2+2ab + b^2$. So, squaring the left side gives $x + 10$, and squaring the right side $(x + 1)^2=x^2+2x + 1$. 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 third option in the given choices).