Solving Quadratic Equations by Square Roots

Learn to solve quadratic equations by taking the square root of both sides.

Intermediate25 minLesson
Printable worksheet

Definition

When a quadratic equation is in the form x2=kx^2 = k, we can solve it by taking the square root of both sides.
x2=k  ⟹  x=±kx^2 = k \implies x = \pm\sqrt{k}
Key concept: Every positive number has TWO square roots:
  • A positive root: k\sqrt{k}
  • A negative root: −k-\sqrt{k}
For example, if x2=16x^2 = 16:
x=±16=±4x = \pm\sqrt{16} = \pm 4
This means x=4x = 4 OR x=−4x = -4 (because both 42=164^2 = 16 and (−4)2=16(-4)^2 = 16).

Try it now

How many solutions does x2=49x^2 = 49 have?

Worked Examples

Solve: x2=25x^2 = 25
1

Identify the equation form

The equation is already in the form x2=kx^2 = k where k=25k = 25 → Ready to take square root
2

Take the square root of both sides

x2=±25\sqrt{x^2} = \pm\sqrt{25} → x=±25x = \pm\sqrt{25}
3

Simplify the square root

25=5\sqrt{25} = 5 → x=±5x = \pm 5
4

Write both solutions

x=5x = 5 or x=−5x = -5 → Two solutions

Common Mistakes

Forgetting the negative solution

Why it's wrong: Students often write only the positive square root, forgetting that (−a)2=a2(-a)^2 = a^2 as well.

Correct: Always write x=±kx = \pm\sqrt{k} to indicate both solutions. For x2=9x^2 = 9, write x=±3x = \pm 3, not just x=3x = 3.

Taking the square root before isolating x2x^2

Why it's wrong: In equations like 2x2=182x^2 = 18, students sometimes try to take 2x2\sqrt{2x^2} directly.

Correct: First divide by 2 to get x2=9x^2 = 9, then take the square root: x=±3x = \pm 3.

Thinking x2=x\sqrt{x^2} = x

Why it's wrong: This is only true for positive xx. For negative xx, x2=∣x∣\sqrt{x^2} = |x|.

Correct: Write x2=±x\sqrt{x^2} = \pm x or more precisely ∣x∣|x|, which gives both positive and negative possibilities.

Attempting to solve x2=−kx^2 = -k (negative) as a real number

Why it's wrong: Students may try to write x=±−kx = \pm\sqrt{-k}, which is not a real number.

Correct: Recognize that if k<0k < 0, the equation x2=kx^2 = k has no real solution.

Watch a video explanation

The same topic explained by another teacher, if a video helps you more.

How to Solve Quadratic Equations Using the Square Root Property

by Understand The Math

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Interactive Visual

Function graph

-8-6-4-202468010203040xy(-5, 25)(5, 25)

Tap or hover over the graph to read off values.

y = x²y = 25
  • Intersection point: (-5, 25)
  • Intersection point: (5, 25)

Equation Solver

Solve for xx:

x2=25x^{2} = 25
1
x2=25x^{2} = 25
Move all terms to one side: ax² + bx + c = 0A quadratic equation is solved from standard form

Tips:

  • • Whatever you do to one side, do to the other
  • • Undo operations in reverse order (PEMDAS backwards)
  • • Check your answer by substituting back

Follow along as we solve the equation step by step.

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Practice Problems

15 problems
Problem 1 of 15
Easy
How many solutions does x2=49x^2 = 49 have?

Why It Matters

The square root method is one of the quickest ways to solve certain quadratic equations:
  • Physics: Finding velocity or distance in equations like v2=2ghv^2 = 2gh
  • Geometry: Calculating side lengths from area (s2=As^2 = A)
  • Engineering: Determining dimensions when area is known
  • Finance: Solving for rates in compound interest formulas
This technique is often faster than factoring or using the quadratic formula, making it essential in your algebra toolkit!

Real World Applications

Calculating Distances

In physics, the relationship between distance, acceleration, and time involves squared terms.

Example:

A ball is dropped and falls according to d=5t2d = 5t^2 (in meters). How long does it take to fall 80 meters? Solve 5t2=805t^2 = 80, so t2=16t^2 = 16, giving t=4t = 4 seconds (we use only the positive value since time cannot be negative).

1Try It Yourself

A stone is dropped from a bridge. The distance fallen is given by d=5t2d = 5t^2 meters.

How long does it take to fall 45 meters?

Step 1: Write the mathematical expression

Set up the equation: 5t2=455t^2 = 45

Finding Side Lengths from Area

When you know the area of a square, you can find the side length using square roots.

Example:

A square garden has an area of 144 square meters. The side length is ss where s2=144s^2 = 144, so s=12s = 12 meters.

2Try It Yourself

A square room has an area of 81 square meters.

What is the length of each wall?

Step 1: Write the mathematical expression

If the side length is ss, then s2=81s^2 = 81

Projectile Motion

The height of a thrown ball involves quadratic equations.

Example:

A ball thrown upward reaches height h=64−16t2h = 64 - 16t^2 feet. When is the ball at 48 feet? Solve 64−16t2=4864 - 16t^2 = 48, giving 16t2=1616t^2 = 16, so t2=1t^2 = 1 and t=1t = 1 second.

3Try It Yourself

A fountain shoots water with height h=100−4t2h = 100 - 4t^2 (in cm, tt in seconds).

At what time is the water at height 36 cm?

Step 1: Write the mathematical expression

Solve: 100−4t2=36100 - 4t^2 = 36

Key Takeaways

  • 1For equations in the form x2=kx^2 = k, take the square root of both sides: x=±kx = \pm\sqrt{k}
  • 2Always include BOTH the positive and negative solutions (±\pm)
  • 3First isolate x2x^2 before taking the square root
  • 4If k<0k < 0, there is no real solution
  • 5This method also works for (ax+b)2=k(ax + b)^2 = k: just solve for the expression inside

Frequently Asked Questions

Because both a positive and negative number, when squared, give the same positive result. For example, 52=255^2 = 25 and (−5)2=25(-5)^2 = 25. So if x2=25x^2 = 25, both x=5x = 5 and x=−5x = -5 are valid solutions.
Because both a positive and negative number, when squared, give the same positive result. For example, 52=255^2 = 25 and (−5)2=25(-5)^2 = 25. So if x2=25x^2 = 25, both x=5x = 5 and x=−5x = -5 are valid solutions.
This method works best when the equation can be written as x2=kx^2 = k or (expression)2=k(\text{expression})^2 = k. If the equation has an xx term (like x2+3x=10x^2 + 3x = 10), you'll need factoring or the quadratic formula instead.
You can leave the answer in radical form (like x=±7x = \pm\sqrt{7}) or approximate with a calculator. Both forms are correct; the radical form is exact.

Glossary

Square root
A number that, when multiplied by itself, gives the original number. 25=5\sqrt{25} = 5 because 5×5=255 \times 5 = 25.
Perfect square
A number that is the square of an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...
Plus-minus (±)
A symbol indicating both positive and negative values. ±3\pm 3 means both +3+3 and −3-3.
Quadratic equation
An equation where the highest power of the variable is 2, such as x2=16x^2 = 16 or x2+5x+6=0x^2 + 5x + 6 = 0.

Formula Card

Square Root Method

x2=k  ⟹  x=±kx^2 = k \implies x = \pm\sqrt{k}

Only valid when $k \geq 0$

General Form

(ax+b)2=k  ⟹  ax+b=±k(ax + b)^2 = k \implies ax + b = \pm\sqrt{k}

Then solve the linear equation

Today's challengeOne problem, three tries

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