Solving Radical Equations

Solving Radical Equations

How to solve radical equations (square root equations): example and its solution.

Example: Solve √x + 6 = x

Find the value of x. Square root [x + 6] = x

Before solving the radical equation,
find the range of x
from the given equation.

For an even root (√, 4, 6, etc.),
the radicand (the number inside the radical)
cannot be (-).

So the radicand of √x + 6, [x + 6],
cannot be (-).

So x + 6 ≥ 0.

Move +6 to the right side.

Then x ≥ -6.

The left side, √x + 6, is not (-).
There's no (-) sign.

So the right side, x, is also not (-).

So x ≥ 0.

x ≥ -6
x ≥ 0

Draw these two inequalities on a number line.
And find the intersection.

Then x ≥ 0.

Next, solve the radical equation.

Square both sides.

(√x + 6)2 = x + 6

Square root - Square of a square root

Move x2 to the left side.

Then -x2 + x + 6 = 0.

Multiply -1 on both sides.

Then x2 - x - 6 = 0.

Factor x2 - x - 6.

Factor a quadratic trinomial

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

The constant term is (-).
So the signs of the numbers are different:
one is (+), and the other is (-).

(-1, 6) and (-2, 3)
are not the right numbers.

[-6] = -3⋅2
-3 + 2 = [-1]
So -3 and 2 are the right numbers.

Use -3 and +2
to write a factored form:
(x - 3)(x + 2) = 0.

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

1) x - 3 = 0
So x = 3.

2) x + 2 = 0
So x = -2.

Solving a quadratic equation by factoring

See if the found x values
are in the initial range of x:
x ≥ 0.

x = 3 is in the range.
But x = -2 is not in the range.

So x = 3 is the answer.