Solving Radical Equations

What You’ll Learn

In this lesson you’ll learn how to solve radical equations by isolating the radical and squaring both sides, while carefully checking for extraneous solutions.

The Concept

A radical equation is an equation that contains a variable inside a radical sign.

General steps to solve:

1. Isolate the radical on one side of the equation.
2. Square both sides to eliminate the radical (or raise to the power of the index).
3. Solve the resulting equation.
4. Check all solutions in the original equation. Some may be extraneous.

Why do extraneous solutions appear? Squaring both sides can introduce extra solutions that weren’t valid in the original equation. For example, squaring turns both 3 and −3 into 9, so you can pick up a “solution” that came from the wrong sign.

Worked Example

1. Solve √(2x + 1) + 3 = 7

Isolate the radical:

$$\sqrt{2x + 1} = 4$$

Square both sides:

$$2x + 1 = 16$$

Solve:

$$2x = 15 \quad\Rightarrow\quad x = 7.5$$

Check: √(2(7.5) + 1) + 3 = √16 + 3 = 4 + 3 = 7 ✓. Valid.

2. Solve √(x + 4) = x − 2 (watch for extraneous solutions)

Square both sides:

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

Bring all terms to one side:

$$0 = x^2 - 5x$$

Factor:

$$x(x - 5) = 0 \quad\Rightarrow\quad x = 0 \;\text{ or }\; x = 5$$

Now check both in the original equation √(x + 4) = x − 2:

• x = 0: √(0 + 4) = √4 = 2, but 0 − 2 = −2. Since 2 ≠ −2, this is extraneous. Throw it out.
• x = 5: √(5 + 4) = √9 = 3, and 5 − 2 = 3. Since 3 = 3, this is valid. ✓

Solution: x = 5 only.

The x = 0 solution was introduced by squaring. It satisfies the squared equation but not the original. This is why checking is essential.

Real-World Application

Radical equations appear in:

• Distance and rate problems (time = √(distance / acceleration))
• Physics formulas (pendulum period, falling objects)
• Engineering (beam strength, pipe flow)
• Finance (certain compound interest models)

Example: the time for an object to fall a certain distance under gravity involves a radical equation.

You’ve Got This

The most important step is checking your solutions in the original equation. Squaring both sides is powerful but can introduce invalid answers. Isolate the radical first, square carefully, solve, then verify. With practice, you’ll spot potential extraneous solutions more quickly.

Quiz

When solving radical equations, why must you check your solutions?

• A.To make sure the answer is positive
• B.Some may be extraneous (don't satisfy the original)
• C.To verify the degree
• D.To find the range

Solve $\sqrt{x + 3} = 5$

• A.$x = 25$
• B.$x = 2$
• C.$x = 8$
• D.$x = 22$

The first step when solving a radical equation is usually to:

• A.Isolate the radical
• B.Square both sides immediately
• C.Factor the expression
• D.Graph the function

If after squaring you get a quadratic, you should:

• A.Ignore the check
• B.Only take the positive root
• C.Check both solutions in the original equation
• D.Only use the quadratic formula

Solve $\sqrt{2x - 1} = x - 2$.

• A.$x = 1$
• B.$x = 5$
• C.$x = 1$ and $x = 5$
• D.No solution