Isolate the radical, then raise both sides to the matching power. A square root needs squaring; a cube root needs cubing.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Isolate first
Squaring before isolating produces cross terms and a worse equation. Get the radical alone first.
Squaring is not reversible
Squaring turns −3 and 3 into the same 9. That is why an answer can satisfy the squared equation and fail the original.
Checking is compulsory
Substitute every candidate into the original. Any that makes a square root equal a negative must be rejected.
Odd roots are safe
Cubing is reversible, so cube root equations do not produce extraneous solutions. Only even powers do.
Two radicals
With two square roots, isolate one, square, then isolate the remaining one and square again.
Squaring can invent solutions
Squaring is not reversible: 3 and −3 both square to 9. So squaring both sides can produce answers that fail the original equation, and every one must be checked.
Isolate the radical first
Squaring a sum containing a root leaves another root behind. Getting the radical alone before squaring is what keeps the process finite, and with two radicals it must be done twice.
Check in the original equation
The squared equation has a larger solution set than the original, so checking against it proves nothing. Substitution into the equation as given is the only valid test.
No solution is a real outcome
If every algebraic candidate fails the check, the equation has no solution. That happens genuinely — a square root cannot equal a negative number — and reporting it is correct.
Step 2: Try It Yourself
Tap and try it out.
- Point(4, 2)
Step 3: Watch an Example
One step at a time.
Watch Kofi Reject a Solution
Kofi solves √(x + 6) = x.
- Step 1
The radical is already isolated, so he squares both sides: x + 6 = x².
Step 4: Your Turn
Practice makes it stick.
The Solve
Problem 1 of 2
√x = 5. What is x?
The Rejection
Problem 2 of 2
√(x + 6) = x. How many valid solutions?
Solve and Check
1 of 8
√x = 7. What is x?
2 of 8
√(x − 3) = 4. What is x?
3 of 8
√x = −2. How many solutions?
4 of 8
The cube root of x = 2. What is x?
5 of 8
Do cube root equations produce extraneous solutions? 1 yes, 0 no.
6 of 8
√(2x) = 6. What is x?
7 of 8
Put the solving process in order.
- 1Raise both sides to the matching power.
- 2Solve the resulting equation.
- 3Check every candidate in the original and reject failures.
- 4Isolate the radical on one side.
8 of 8
√(x + 1) = 3. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
√(x − 2) = 5. What is x?
Question 2 of 2
Why can squaring introduce extraneous solutions?
What You Learned
- Isolate the radical, then raise both sides to the matching power.
- Squaring is not reversible, so extraneous solutions can appear.
- Check every candidate in the original equation.