Question

Solve the equation for 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 equation is

$\sqrt{x+6}-4=x$

Add 4 to both sides of the equation.

$\sqrt{x+6}=x+4$

Square both sides

$x+6=(x+4)^2$

$x+6=x^2+8x+16$

Rewrite in standard form;

$x^2+8x-x+16-6=0$

$x^2+7x+10=0$

$x^2+7x+10=0$

$(x+2)(x+5)=0$

x=-2 or x=-5

Checking for extraneous solution.

When x=-2

$\sqrt{-2+6}-4=-2$

$\sqrt{4}-4=-2$

$2-4=-2$. This statement is true. This implies that: x=-2 is a solution.

When x=-5

$\sqrt{-5+6}-4=-5$

$\sqrt{1}-4=-5$

$1-4=-5$. This statement is not true. This implies that: x=-5 is an extranous solution.