How to Solve Radical Equations

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Radical equations are those that contain radical expressions. Applying exponentiation rules and some basic algebraic laws are necessary for solving radical equations.

Solving Radical Equations

A simple way to solve radical equations is to first isolate the radical term and then raise both sides to a power to get rid of the radical. This is the same way you solved other equations that didn't have radicals: you rearrange the expression to find the variable you want to know, and then you solve the equation that's left.

To solve radical equations, you will need to keep in mind two main ideas.

The first one is if \(a \ = \ b\), \(a^2 \ = \ b^2\). (Because of this property, you can square both sides of an equation and be sure that they are still equal.)
The second thing is that if you square the square root of any positive number \(x\), you get \(x\): \((\sqrt{x})^2 \ = \ x\). (With this property, you can get rid of the radicals in your equations.)

Let's start with a radical equation that's easy to solve: \(\sqrt{x} \ + \ 4 \ = \ 6\)

Step1: Minus \(4\) from both sides of the equation to isolate the variable term on the left side.
\(\sqrt{x} \ + \ 4 \ = \ 6 \ ⇒ \ \sqrt{x} \ = \ 2\)

Step2: Square both sides to get rid of the radical. Then simplify.
\(\sqrt{x} \ = \ 2 \ ⇒ \ (\sqrt{x})^2 \ = \ 4 \ ⇒ \ x \ = \ 4\)

In the example above, the only thing that was under the radical was the variable \(x\). Sometimes you'll need to figure out how to solve an equation that has more than one term under a radical. Follow the same steps to solve these, but pay attention to a key point: square both sides of an equation, not the individual terms. See how the next example is solved.

Example

Solve for \(x\): \(\sqrt{x \ - \ 2} \ + \ 6 \ = \ 10\)

Solution:

Step1: Isolate the variable terms on one side, to do that minus \(6\) from both sides:
\(\sqrt{x \ - \ 2} \ + \ 6 \ = \ 10 \ ⇒ \ \sqrt{x \ - \ 2} \ = \ 4\)

Step2: To remove the radical, square both sides:
\((\sqrt{x \ - \ 2})^2 \ = \ 4^2 \ ⇒ \ x \ - \ 2 \ = \ 16\)

Step3: Now, make the equation as simple as possible and find \(x\):
\(x \ - \ 2 \ = \ 16 \ ⇒ \ x \ = \ 18\)

Step4: Check your answer. If you replace \(x\) in the original equation with \(18\), you get a true statement, so the solution is right:
\(\sqrt{x \ - \ 2} \ + \ 6 \ = \ 10 \ ⇒ \ \sqrt{18 \ - \ 2} \ + \ 6 \ = \ 10 \ ⇒ \ \sqrt{16} \ + \ 6 \ = \ 10 \ ⇒ \ 4 \ + \ 6 \ = \ 10\)

Solving Radical Equations

Think of this lesson as more than a rule to memorize. Solving Radical Equations is about roots, simplification, restrictions, and extraneous solutions. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Radicals undo powers. For square roots, \(\sqrt{a}\) means the nonnegative number whose square is \(a\), and perfect-square factors help simplify expressions.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

Clear clutter such as parentheses or fractions.
Collect like terms.
Undo operations in reverse order.
Substitute the answer back or test a point.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Free printable Worksheets

Exercises for Solving Radical Equations

1) Solve for \(x\): \(\sqrt{x \ + \ 5} \ = \ 4\)

2) Solve for \(x\): \(\sqrt{2x \ - \ 1} \ = \ 5\)

3) Solve for \(x\): \(\sqrt{x \ - \ 3} \ + \ 2 \ = \ 7\)

4) Solve for \(x\): \(3\sqrt{x \ + \ 1} \ = \ 12\)

5) Solve for \(x\): \(\sqrt{x \ + \ 6} \ = \ x\)

6) Solve for \(x\): \(\sqrt{2x \ + \ 9} \ = \ x \ + \ 3\)

7) Solve for \(x\): \(\sqrt{5x \ + \ 1} \ = \ \sqrt{2x \ + \ 10}\)

8) Solve for \(x\): \(\sqrt{x \ + \ 10} \ - \ \sqrt{x \ + \ 1} \ = \ 1\)

9) Solve for \(x\): \(\sqrt{3x \ + \ 4} \ = \ x\)

10) Solve for \(x\): \(\sqrt{x \ + \ 7} \ = \ x \ - \ 1\)

11) Solve for \(x\): \(\sqrt{2x \ - \ 3} \ + \ 1 \ = \ x\)

12) Solve for \(x\): \(\sqrt{x \ + \ 4} \ + \ \sqrt{x \ - \ 1} \ = \ 5\)

13) Solve for \(x\): \(\sqrt{4x \ + \ 9} \ = \ 2x \ - \ 3\)

14) Solve for \(x\): \(\sqrt{x^2 \ + \ 5} \ = \ x \ + \ 1\)

15) Solve for \(x\): \(\sqrt{7x \ - \ 6} \ = \ x\)

16) Solve for \(x\): \(\sqrt{x \ + \ 6} \ + \ \sqrt{x \ - \ 3} \ = \ 9\)

17) Solve for \(x\): \(\sqrt{x \ + \ 2} \ - \ \sqrt{x \ - \ 7} \ = \ 3\)

18) Solve for \(x\): \(\sqrt{x \ + \ 14} \ = \ x \ - \ 2\)

19) Solve for \(x\): \(\sqrt{x \ + \ 5} \ + \ \sqrt{x} \ = \ 5\)

20) Solve for \(x\): \(\sqrt{x \ + \ 1} \ = \ x \ - \ 5\)

Answers

1)Solve for \(x\): \(\sqrt{x \ + \ 5} \ = \ 4\)