Radical Equation

Before solving the equation,
first find the conditions from each x.

See √x + 6.

The number inside the square root
should be (+) (or 0).

So x + 6 ≥ 0.

Move +6 to the right side.

Then x ≥ -6.

Next, find the condition
from the right side x.

The left side, √x + 6, is (+).

So the right side, x,
is also (+).

So x ≥ 0.

x ≥ -6
x ≥ 0

Draw the inequalities on a number line.

x should satisfy both conditions.
So color the common region.

The common region is
x ≥ 0.

So x should satisfy
x ≥ 0.

This is the condition for x.

Next, solve the given radical equation
√x + 6 = x.

Square both sides.

Then
x + 6 = x².

Move x² to the left side.

Multiply -1 to both sides.

Factor the right side
x² - x - 6.

Find a pair of numbers
whose product is the constant term -6
and whose sum is the coefficient of the middle term -1.

-3⋅2 = -6
-3 + 2 = -1

Then (x - 3)(x + 2) = 0.

Factor a Quadratic Trinomial

Solve the quadratic equation.

Case 1) x - 3 = 0
Then x = 3.

Draw x = 3
on the upper number line.

x = 3 is in the colored region.

So x = 3 is the answer for case 1.

Case 2) x + 2 = 0
Then x = -2.

Draw x = -2
on the upper number line.

x = -2 is not in the colored region.

So x = -2 is not the answer.

Case 1) x = 3
Case 2) No root

So x = 3.

So x = 3 is the answer.