Solving Radical Equations

Analyze and transform radical functions including domain/range. Math 30-1.

Lesson 6.1 of Radical & Rational Functions in Math 30-1 — Alberta curriculum lessons.

What you'll learn in this lesson

What Are Radical Equations?. Discover what makes radical equations different from regular equations. Learn the three golden rules for solving them: state restrictions first, isolate then square, and always check for extraneous roots. Understand why squaring both sides can create fake solutions and how to catch them.
Algebraic Method — Worked Examples. Work through two detailed examples step-by-step. See how to state restrictions, isolate radicals, square both sides, and verify solutions. Learn how to identify extraneous roots through Example 1b, which demonstrates why checking restrictions is critical.
Two Variables & Graphical Method. Master advanced radical equations with two variables requiring double squaring. Discover two powerful graphical methods: finding x-intercepts of y = f(x) or finding intersections of y = f(x) and y = g(x). See both approaches visualized with interactive graphs.
Special Cases, More Examples & Practice. Explore special cases like finding x-intercepts and equations with no solution. Learn to recognize when a radical equation is impossible algebraically and see visual proof with graphs. Master the complete skill set with challenging practice questions.

Definition. A radical equation is any equation where the variable appears inside a square root (or other radical).

For example: √(12 - x) + 4 = 7 is a radical equation because x is trapped inside the square root.

The Challenge. You can't just "move things around" like a normal equation — the square root gets in the way.

The big idea is: square both sides to eliminate the radical.

But this creates a risk: squaring can produce fake answers called extraneous roots, so you must always check your answer.

⚠️ Remember: you are squaring sides, NOT terms!. (√(12-x) + 4)^2 ≠ (12-x) + 16 ← this is WRONG

You must isolate the radical first.

Question. State any restrictions, then algebraically find all roots.

Step 1 — State the Restriction. The expression under the square root must be ≥ 0:

12 - x ≥ 0

-x ≥ -12

x ≤ 12

So x must be 12 or less.

Step 2 — Isolate the Radical. Subtract 4 from both sides:

7 - 4 = √(12 - x)

3 = √(12 - x)

The radical is now alone on the right side.

Step 3 — Square Both Sides. (3)^2 = (√(12 - x))^2

9 = 12 - x

Notice: because the radical was isolated first, squaring it simply removes it cleanly.

Step 4 — Solve for x. 9 - 12 = -x

-3 = -x

x = 3

Step 5 — Check the Restriction. Is x = 3 ≤ 12? Yes ✓

Step 6 — Verify in the Original Equation. 7 = √(12 - 3) + 4

7 = √(9) + 4

7 = 3 + 4

7 = 7 ✓

Complete Summary. Original equation: 7 = √(12 - x) + 4

Restriction: x ≤ 12

Solution: x = 3

Verified: both sides equal 7 ✓

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