Name: Radical equations and extraneous solutions: solution walkthrough
Uploaded: 2026-11-13
Duration: 2 min 19 s
Description: What is the solution set of the equation √x + 7 = x - 5? Try it, then watch the animated solution.

A solution is an

x

where the curve

y = \sqrt{x + 7}

meets the line

y = x - 5

. To find it, remove the root by squaring.

Square both sides:

x + 7 = (x - 5)^2

. Multiply out the right side:

x + 7 = x^2 - 10x + 25

.

Move everything to one side:

0 = x^2 - 11x + 18 = (x - 2)(x - 9)

. So

x = 2

or

x = 9

.

Check both in the original equation.

x = 9

:

\sqrt{16} = 4

and

9 - 5 = 4

.

x = 2

:

\sqrt{9} = 3

but

2 - 5 = {-3}

.

Squaring hides the sign:

3^2 = ({-3})^2

. So

x = 2

fits the mirror curve, not ours. Only

x = 9

is a solution.

Traps: keeping both values without checking, keeping the wrong one, or a sign error when factoring.

Try it first