Radical Expressions • Topic 4 of 4

Radical Equations

To solve an equation with a square root, isolate the radical, then square both sides to remove it, and solve the resulting equation. Because squaring can introduce extraneous solutions, always check each answer in the original equation. For √(x + 3) = 5, square to get x + 3 = 25, so x = 22. If a constant is added to the radical, move it first: √x + 4 = 9 gives √x = 5, then x = 25. Radical equations appear on the SAT in geometry (Pythagorean) and Advanced Math contexts, where the check step is essential to discard false roots.

✅ Solved examples

1. Solve √x = 6.
Square both sides: x = 36.
2. Solve √(x + 3) = 5.
Square: x + 3 = 25, so x = 22.
3. Solve √x + 4 = 9.
Isolate: √x = 5; square: x = 25.
4. Solve √(2x − 1) = 3.
Square: 2x − 1 = 9, so 2x = 10 and x = 5.

✏️ Practice — try these, take hints as needed

1. Solve √x = 10.
Square both sides.
10².
—
100.
2. Solve √(x − 2) = 4.
Square: x − 2 = 16.
Add 2.
—
18.
3. Solve √x − 3 = 2.
Isolate: √x = 5.
Square.
—
25.
4. Solve √(3x) = 6.
Square: 3x = 36.
Divide by 3.
—
12.
5. Solve √(x + 1) = 7.
Square: x + 1 = 49.
Subtract 1.
—
48.

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Rationalizing Denominators Take the chapter test →