Question

Solve the equation x. If a solution is extraneous, be sure to identify it in your final answer.

Answer 1

Answer:

x=2 is the only solution

Step-by-step explanation:

The given expression is;

$\sqrt{x+7}-1=x$

Add 1 to both sides of the equation;

$\sqrt{x+7}=x+1$

Square both sides ;

$(\sqrt{x+7})^2=(x+1)^2$

$x+7=x^2+2x+1$

This implies that;

$x^2+2x-x+1-7=0$

$x^2+x-6=0$

Split the middle term

$x^2+3x-2x-6=0$

$x(x+3)-2(x+3)=0$

$(x+3)(x-2)=0$

x=-3,or x=2

Check for extraneous solutions by substituting into the original equation.

When x=-3,

$\sqrt{-3+7}-1=-3$

$\sqrt{4}-1\ne-2$

This is an extraneous solution.

When x=2,

$\sqrt{2+7}-1=2$

$\sqrt{9}-1=2$

$3-1=2$. This is true

This is not an extraneous solution