Solving Quadratic Equations by Square Roots

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

Intermediate25 minLesson

Definition

When a quadratic equation is in the form

x^2 = k

, we can solve it by taking the square root of both sides.

x^2 = k \implies x = \pm\sqrt{k}

Key concept: Every positive number has TWO square roots:

A positive root: $\sqrt{k}$
A negative root: $-\sqrt{k}$

For example, if

x^2 = 16

x = \pm\sqrt{16} = \pm 4

This means

x = 4

x = -4

(because both

4^2 = 16

and

(-4)^2 = 16

Try it now

How many solutions does

x^2 = 49

have?

Worked Examples

Solve: $x^2 = 25$

Identify the equation form

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

Take the square root of both sides

$\sqrt{x^2} = \pm\sqrt{25}$ → $x = \pm\sqrt{25}$

Simplify the square root

$\sqrt{25} = 5$ → $x = \pm 5$

Write both solutions

$x = 5$ or $x = -5$ → Two solutions

Answer: $x = 5$ or $x = -5$

Common Mistakes

Forgetting the negative solution

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

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

Taking the square root before isolating $x^2$

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

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

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

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

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

Attempting to solve $x^2 = -k$ (negative) as a real number

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

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

Interactive Visual

Balance Scale

x + 3=7

Apply to both sides:

Solution: x = 4

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Interactive Sandbox

Expression Calculator

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History

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Practice Problems

15 problems

Problem 1 of 15

Easy

How many solutions does $x^2 = 49$ 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 $v^2 = 2gh$
Geometry: Calculating side lengths from area ( $s^2 = A$ )
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 $d = 5t^2$ (in meters). How long does it take to fall 80 meters? Solve $5t^2 = 80$ , so $t^2 = 16$ , giving $t = 4$ 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 $d = 5t^2$ meters.

How long does it take to fall 45 meters?

Step 1: Write the mathematical expression

Set up the equation: $5t^2 = 45$

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 $s$ where $s^2 = 144$ , so $s = 12$ 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 $s$ , then $s^2 = 81$

Projectile Motion

The height of a thrown ball involves quadratic equations.

Example:

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

3Try It Yourself

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

At what time is the water at height 36 cm?

Step 1: Write the mathematical expression

Solve: $100 - 4t^2 = 36$

Key Takeaways

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

5^2 = 25

and

(-5)^2 = 25

. So if

x^2 = 25

, both

x = 5

and

x = -5

are valid solutions.

Glossary

Square root: A number that, when multiplied by itself, gives the original number. $\sqrt{25} = 5$ because $5 \times 5 = 25$ .
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. $\pm 3$ means both $+3$ and $-3$ .
Quadratic equation: An equation where the highest power of the variable is 2, such as $x^2 = 16$ or $x^2 + 5x + 6 = 0$ .

Formula Card

Square Root Method

x^2 = k \implies x = \pm\sqrt{k}

Only valid when $k \geq 0$

General Form

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

Then solve the linear equation

Solving Quadratic Equations by Square Roots

Definition

Worked Examples

Common Mistakes

Interactive Visual

Balance Scale

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Calculating Distances

Finding Side Lengths from Area

Projectile Motion

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Square Root Method

General Form

More in This Topic

Introduction to Quadratic Equations

Vertex Form

Completing the Square

The Quadratic Formula