Virginia SOL Math / Algebra II / Equations & Inequalities

A2.EI.5

Equations With Radical Expressions

What you'll learn

  • Solve equations containing a square root
  • Isolate the radical, then square both sides
  • Check for and reject extraneous solutions

Words to know

Radical equation —
an equation with a variable under a radical
Isolate —
get the radical alone on one side
Square both sides —
raise both sides to the second power
Extraneous solution —
a value that fails in the original equation
Check —
substitute back to confirm a solution

Why it matters Formulas for distance, period, and speed often involve square roots, so solving radical equations recovers the original quantity.

Worked examples

Model ItI do

Solve sqrt(x + 3) = x - 3.

The radical is already isolated, so square both sides: x + 3 = (x - 3)^2 = x^2 - 6x + 9. Rearrange: x^2 - 7x + 6 = 0, which factors as (x - 1)(x - 6) = 0. Testing shows x = 6 works but x = 1 does not, so x = 6.

Try It TogetherWe do

Solve sqrt(x) = 5.

Square both sides: x = 25. Check: sqrt(25) = 5, which is true, so x = 25.

Now YouYou do

Solve sqrt(x - 2) = 4.

Square both sides, then solve for x and check. Check your answer in the back of this chapter.

Practice

Mastered 0 of 11

Warm-up

Ease in with the basics.

  1. 1.Solve sqrt(x) = 9.

  2. 2.Solve sqrt(x + 5) = 4.

  3. 3.Solve sqrt(3x) = 9.

On Level

The core Quick Check from the book.

  1. 1.Solve sqrt(x) = 7.

  2. 2.Solve sqrt(x + 1) = 3.

  3. 3.Solve sqrt(2x) = 6.

  4. 4.Why must radical solutions be checked?

  5. 5.Solve sqrt(x - 5) = 0.

Challenge

Stretch your thinking.

  1. 1.Solve sqrt(x + 7) = x - 5.

  2. 2.Solve sqrt(2x + 3) = x.

  3. 3.Solve sqrt(x - 1) + 3 = x.