Eliminating Square Roots from Equations

When you first learned about squared numbers like 32, 52 and ​x​2, you probably learned about a squared number’s inverse operation, the square root, too. That inverse relationship between squaring numbers and square roots is important, because in plain English it means that one operation undoes the effects of the other. That means that if you have an equation with square roots in it, you can use the “squaring” operation, or exponents, to remove the square roots. But there are some rules about how to do this, along with the potential trap of false solutions.

TL;DR (Too Long; Didn’t Read)

To solve an equation with a square root in it, first isolate the square root on one side of the equation. Then square both sides of the equation and continue solving for the variable. Don’t forget to check your work at the end.

A Simple Example

Before considering some of the potential “traps” of solving an equation with square roots in it, consider a simple example: Solve the following equation for ​x​:

(sqrt{x} + 1 = 5)

1. Isolate the Square Root

Use arithmetic operations like addition, subtraction, multiplication and division to isolate the square root expression on one side of the equation. For example, if your original equation was √​x​ + 1 = 5, you would subtract 1 from both sides of the equation to get the following:

(sqrt{x} = 4)

2. Square Both Sides of the Equation

Squaring both sides of the equation eliminates the square root sign. This gives you:

((sqrt{x})^2 = (4)^2)

Or, once simplified:

(x = 16)

You’ve eliminated the square root sign ​and​ you have a value for ​x​, so your work here is done. But wait, there’s one more step:

3. Check Your Work

Check your work by substituting the ​x​ value you found into the original equation:

(sqrt{16} + 1 = 5)

Next, simplify:

(4 + 1 = 5)

And finally:

(5 = 5)

Because this returned a valid statement (5 = 5, as opposed to an invalid statement like 3 = 4 or 2 = -2, the solution you found in Step 2 is valid. In this example, checking your work seems trivial. But this method of eliminating radicals can sometimes create “false” answers that don’t work in the original equation. So it’s best to get in the habit of always checking your answers to make sure they return a valid result, starting now.

A Slightly Harder Example

What if you have a more complex expression underneath the radical (square root) sign? Consider the following equation. You can still apply the same process used in the previous example, but this equation highlights a couple of rules you must follow.

(sqrt{y – 4} + 5 = 29)

1. Isolate the Radical

As before, use operations like addition, subtraction, multiplication and division to isolate the radical expression on one side of the equation. In this case, subtracting 5 from both sides gives you:

(sqrt{y – 4} = 24)

Warning

Note that you’re being asked to isolate the square root (which presumably contains a variable, because if it was a constant like √9, you could just solve it on the spot; √9 = 3). You are ​_not_​ being asked to isolate the variable. That step comes later, after you’ve eliminated the square root sign.

2. Square Both Sides

Square both sides of the equation, which gives you the following:

({sqrt{y – 4})^2 = (24)^2)

Which simplifies to:

(y – 4 = 576)

Warning

Note that you must square everything underneath the radical sign, not just the variable.

3. Isolate the Variable

Now that you’ve eliminated the radical or square root from the equation, you can isolate the variable. To continue the example, adding 4 to both sides of the equation gives you:

(y = 580)

4. Check Your Work

As before, check your work by substituting the ​y​ value you found back into the original equation. This gives you:

(sqrt{580 – 4} + 5 = 29)

Which simplifies to:

(sqrt{576} + 5 = 29)

Simplifying the radical gives you:

(24 + 5 = 29)

And finally:

(29 = 29)

a true statement that indicates a valid result.