Math · College algebra · Concept
Solving radical equations
To solve an equation with a square root, isolate the root, square both sides and solve. Squaring can create answers that do not work, so substitute every candidate into the original equation and keep only those that satisfy it.
Isolate the radical first
Get the square root alone on one side before squaring. Squaring √(x + 7) + 5 as it stands would produce a cross term that still contains the root; squaring √(x + 7) = x − 5 removes the root completely.
Square both sides
If two numbers are equal, their squares are equal, so squaring keeps every solution. The right side is a binomial: (x − 5)² = x² − 10x + 25, not x² − 25.
Why squaring can add a solution
The converse fails: equal squares only mean the numbers are equal or opposite. √(x + 7) is never negative, but x − 5 can be. A value where √(x + 7) = −(x − 5) also satisfies the squared equation, and it is extraneous.
Can you tell in advance which candidate fails?
Yes. Since √(x + 7) ≥ 0, a solution must also make x − 5 ≥ 0, that is x ≥ 5, so any candidate below 5 fails. The check in the original equation settles it either way.
Check every candidate in the original equation
Substitute each candidate into the original equation, not the squared one: the squared equation accepts both. Keep only the values that make the original true.
Two radicals, or a different root
With two square roots, isolate one, square, then isolate the remaining root and square again, and check at the end. A cube root is cleared by cubing, which is reversible for real numbers, so it does not add extraneous solutions.
Common mistakes
- Squaring before isolating the radical, which leaves a root in the cross term.
- Squaring a binomial term by term: (x − 5)² is x² − 10x + 25, not x² − 25.
- Checking candidates in the squared equation instead of the original, which accepts the extraneous value.
- Rejecting a candidate only because it is negative: the check in the original equation decides, not the sign of x.
Key terms
- Radical equation
- An equation with the variable under a root. Isolating the root and raising both sides to a power can introduce extraneous solutions, so every candidate is checked in the original equation.
- Extraneous solution
- A value found while solving that doesn’t satisfy the original equation. Squaring both sides or multiplying by an expression with the variable can create one, so check answers in the original.
- Principal root
- The one root a radical sign stands for: √x means the nonnegative square root, so √9 = 3, not −3. Solving x² = 9 still gives both 3 and −3.
- nth root
- A number whose nth power is the given value. An odd root, such as ∛(−8) = −2, exists for every real number; in the real numbers an even root needs a nonnegative number.
- Domain
- The set of inputs for which a function or expression is defined. In the real numbers that rules out zero denominators, negative numbers under even roots and inputs of logarithms that aren’t positive.
Work through an example
Solve √(x + 7) + 5 = x.
Solve a radical equation and check the roots →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Check the steps in Math Graph both sides in Graph Open worked example on a board Checking after squaring in Math ReferenceYour existing work stays on this device. Examples open as editable copies.