Solving Equations Using the Square Root Property

What Is the Square Root Property?

The Square Root Property is a straightforward technique for solving quadratic equations. If you have an equation in the form x² = c, you can solve it by taking the square root of both sides.

That's it. That's the whole property.

It works because if x² = c, then x = ±√c. The ± symbol matters—you're not ignoring it. Every perfect square has two square roots, one positive and one negative.

When to Use This Method

The square root property works best when:

• Your equation already has a squared variable isolated on one side
• No x-term exists (only x² and a constant)
• The constant on the right side is a perfect square—or you're fine working with irrational numbers

If your equation has an x-term, you'll need to rearrange it first or use a different method like completing the square or the quadratic formula.

How to Apply the Square Root Property

Step-by-Step Process

1. Isolate the squared term — Get x² alone on one side of the equation
2. Take the square root of both sides
3. Add the ± symbol on the right side
4. Solve for x — you'll get two solutions

Simple Example

Let's solve x² = 25.

Take the square root of both sides:

x = ±√25

x = ±5

So x = 5 or x = -5.

You can verify: 5² = 25 and (-5)² = 25. Both work.

Example with Fractions

Solve x² = 9/16.

x = ±√(9/16)

x = ±(3/4)

Check: (3/4)² = 9/16 ✓ and (-3/4)² = 9/16 ✓

Example with Irrational Solutions

Solve x² = 7.

x = ±√7

Your solutions are x = √7 and x = -√7. That's fine. You don't need to simplify √7 further—it doesn't simplify nicely.

Approximate values are x ≈ 2.646 and x ≈ -2.646 if you need decimals.

Example with a Coefficient

Solve 4x² = 36.

First, divide both sides by 4:

x² = 9

Now apply the square root property:

x = ±3

Solving Equations with (ax + b)² = c

Sometimes the squared term includes a coefficient and constant inside the parentheses.

Solve (x - 3)² = 16.

Take the square root of both sides:

x - 3 = ±√16

x - 3 = ±4

Now split into two equations:

x - 3 = 4 → x = 7

x - 3 = -4 → x = -1

Solutions: x = 7 or x = -1.

Check: (7-3)² = 16 ✓ and (-1-3)² = 16 ✓

Common Mistakes to Avoid

• Forgetting the ± — This is the most common error. You always need both the positive and negative root unless the problem specifies otherwise
• Taking the square root of only one term — You must take the square root of the entire side
• Not isolating first — If x² isn't alone, handle the coefficient before taking roots
• Sign errors inside parentheses — Double-check your distribution before squaring

Square Root Property vs. Other Methods

Here's how the square root property compares to other solving techniques:

The square root property is the fastest method when your equation fits the pattern. Use it whenever possible instead of grinding through the quadratic formula.

Practice Problems

Try these on your own before checking answers:

1. x² = 49 → Answer: x = ±7
2. x² = 12 → Answer: x = ±2√3
3. 9x² = 81 → Answer: x = ±3
4. (x + 2)² = 25 → Answer: x = 3 or x = -7
5. (2x - 1)² = 49 → Answer: x = 4 or x = -3

When This Method Falls Short

The square root property only works when your variable is already squared. If you have an equation like x² + 6x + 5 = 0, you can't just take square roots.

For those cases, you have options:

• Factor the equation first
• Complete the square
• Apply the quadratic formula

Each method has its place. Knowing which one to use comes down to recognizing the equation's form.

Quick Reference

The Rule: If x² = c, then x = ±√c

Steps:

1. Isolate x²
2. Square root both sides
3. Include ±
4. Solve

Keep this process in your toolkit. The square root property won't solve every quadratic you'll encounter, but when it does apply, it's the fastest path to the answer.