solve math problems square roots

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Education and Communications
Mathematics

How to Solve Square Root Problems

Last Updated: November 3, 2022 References

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been viewed 334,866 times.

While the intimidating sight of a square root symbol may make the mathematically-challenged cringe, square root problems are not as hard to solve as they may first seem. Simple square root problems can often be solved as easily as basic multiplication and division problems. More complex square root problems, on the other hand, can require some work, but with the right approach, even these can be easy. Start practicing square root problems today to learn this radical new math skill!

Understanding Squares and Square Roots

Try squaring a few more numbers on your own to test this concept out. Remember, squaring a number is just multiplying it by itself. You can even do this for negative numbers. If you do, the answer will always be positive. For example, (-8) 2 = -8 × -8 = 64 .

Image titled Solve Square Root Problems Step 2

Image titled Solve Square Root Problems Step 3

On the other hand, numbers that don't give whole numbers when you take their square roots are called imperfect squares . When you take one of these numbers' square roots, you usually get a decimal or fraction. Sometimes, the decimals involved can be quite messy. For instance, √(13) = 3.605551275464...

Image titled Solve Square Root Problems Step 4

Image titled Solve Square Root Problems Step 5

Image titled Solve Square Root Problems Step 6

Note that although imaginary numbers can't be represented with ordinary digits, they can still be treated like ordinary numbers in many ways. For instance, the square roots of negative numbers can be squared to give those negative numbers, just like any other square root. For example, i 2 = -1

Using Long Division-Style Algorithms

Image titled Solve Square Root Problems Step 7

Image titled Solve Square Root Problems Step 8

In our example, we would divide 6.45 into pairs like this: 6-.45-00 . Note that there is a "leftover" digit on the left — this is OK.

Image titled Solve Square Root Problems Step 9

In our example, the first group in 6-.45-00 is 6. The biggest number that is less than or equal to 6 when squared is 2 — 2 2 = 4. Write a "2" above the 6 under the radical.

Image titled Solve Square Root Problems Step 10

In our example, we would start by taking the double of 2, the first digit of our answer. 2 × 2 = 4. Next, we would subtract 4 from 6 (our first "group"), getting 2 as our answer. Next, we would drop down the next group (45) to get 245. Finally, we would write 4 once more to the left, leaving a small space to add onto the end, like this: 4_.

Image titled Solve Square Root Problems Step 11

In our example, we want to find the number to fill in the blank in 4_ × _ that makes the answer as large as possible but still less than or equal to 245. In this case, the answer is 5 . 45 × 5 = 225, while 46 × 6 = 276.

Image titled Solve Square Root Problems Step 12

Continuing from our example, we would subtract 225 from 245 to get 20. Next, we would drop down the next pair of digits, 00, to make 2000. Doubling the numbers above the radical sign, we get 25 × 2 = 50. Solving for the blank in 50_ × _ =/< 2,000, we get 3 . At this point, we have "253" above the radical sign — repeating this process once again, we get a 9 as our next digit.

Image titled Solve Square Root Problems Step 13

In our example, the number under the radical sign is 6.45, so we would simply slide the point up and place it between the 2 and 5 digits of our answer, giving us 2.539 .

Quickly Estimating Imperfect Squares

Image titled Solve Square Root Problems Step 14

For example, let's say we need to find the square root of 40. Since we've memorized our perfect squares, we can say that 40 is in between 6 2 and 7 2 , or 36 and 49. Since 40 is greater than 6 2 , its square root will be greater than 6, and since it is less than 7 2 , its square root will be less than 7. 40 is a little closer to 36 than it is to 49, so the answer will probably be a little closer to 6. In the next few steps, we'll narrow our answer down.

Image titled Solve Square Root Problems Step 15

In our example problem, a reasonable estimate for the square root of 40 might be 6.4 , since we know from above that the answer is probably a little closer to 6 than it is to 7.

Image titled Solve Square Root Problems Step 16

Image titled Solve Square Root Problems Step 17

In our example, let's pick 6.33 for our two-decimal point estimate. Multiply 6.33 by itself to get 6.33 × 6.33 = 40.0689. Since this is slightly above our original number, we'll try a slightly lower number, like 6.32. 6.32 × 6.32 = 39.9424. This is slightly below our original number, so we know that the exact square root is between 6.33 and 6.32 . If we wanted to continue, we would keep using this same approach to get an answer that's continually more and more accurate.

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For quick solutions, use a calculator. Most modern calculators can instantly find square roots. Usually, all you need to do is to simply type in your number, then press the button with the square root symbol. To find the square root of 841, for example, you might press: 8, 4, 1, (√) and get an answer of 29 . [16] X Research source ⧼thumbs_response⧽ Helpful 0 Not Helpful 0

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Find the Maximum or Minimum Value of a Quadratic Function Easily

About This Article

To solve square root problems, understand that you are finding the number that, when multiplied by itself, equals the number in the square root. For quick recall, memorize the first 10-12 perfect squares, so that you recognize the square root of numbers like 9, 25, 49, or 121. If possible, break the number under the square root into individual perfect squares. For example, √(900) can be broken into √(9) × √(100), and √(100) can be broken into √(25) × √(4), reducing the problem to √(9) × √(25) × √(4), or 3 x 5 x 2 for an answer of 30. If you want to learn how to estimate imperfect square roots, keep reading the article! Did this summary help you? Yes No

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Introduction

Practice question 1: isolating the radical term.

Practice question 2: Two possible solutions

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Calculator Soup ®

Online Calculators

Calculator Use

Use this calculator to find the principal square root and roots of real numbers. Inputs for the radicand x can be positive or negative real numbers. The answer will also tell you if you entered a perfect square.

The answer will show you the complex or imaginary solutions for square roots of negative real numbers. See also the Simplify Radical Expressions Calculator to simplify radicals instead of finding fractional (decimal) answers.

Square Roots, odd and even:

There are 2 possible roots for any positive real number. A positive root and a negative root. Given a number x , the square root of x is a number a such that a 2 = x . Square roots is a specialized form of our common roots calculator .

"Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3, since (-3) 2 = (+3) 2 = 9. Any nonnegative real number x has a unique nonnegative square root r; this is called the principal square root .......... For example, the principal square root of 9 is sqrt(9) = +3, while the other square root of 9 is -sqrt(9) = -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root."[1].

Perfect Square Calculator

This calculator will also tell you if the number you entered is a perfect square or is not a perfect square. A perfect square is a number x where the square root of x is a number a such that a 2 = x and a is an integer. For example, 4, 9 and 16 are perfect squares since their square roots, 2, 3 and 4, respectively, are integers.

Example Square Roots:

To calculate fractional exponents use our calculator for Fractional Exponents .

[1] Weisstein, Eric W. "Square Root." From MathWorld -- A Wolfram Web Resource. Square Root

Square Roots Calculator

Find square roots of any number step-by-step.

Order of Operations

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9.6: Solve Equations with Square Roots

Learning Objectives

By the end of this section, you will be able to:

Before you get started, take this readiness quiz.

Solve Radical Equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.

Definition: RADICAL EQUATION

An equation in which the variable is in the radicand of a square root is called a radical equation .

As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.

But remember that when we write $\sqrt{a}$ we mean the principal square root. So $\sqrt{a} \ge 0$ always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make $\sqrt{a}$ negative. This algebraic solution would not be a solution to the original radical equation ; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.

Example $\PageIndex{1}$

For the equation $\sqrt{x+2}=x$:

1. Is x=2 a solution?

2. Is x=−1 a solution?

Example $\PageIndex{3}$

For the equation $\sqrt{−x+2}=x$:

Is x=1 a solution?

For$a \ge 0$, $(\sqrt{a})^2=a$

How to Solve Radical Equations

$This table has three columns and four rows. The first row says, “Step 1. Isolate the radical on one side of equation. The square root of (2x minus 1) is already isolated on the left side.” It then shows the equation: the square root of (2x minus 1) equals 7.$

Example $\PageIndex{5}$

Solve: $\sqrt{3x−5}=5$.

Example $\PageIndex{6}$

Solve: $\sqrt{4x+8}=6$.

Definition: SOLVE A RADICAL EQUATION.

Example $\PageIndex{7}$

Solve: $\sqrt{5n−4}−9=0$.

Example $\PageIndex{8}$

Solve: $\sqrt{3m+2}−5=0$.

$\frac{23}{3}$

Example $\PageIndex{9}$

Solve: $\sqrt{10z+1}−2=0$.

$\frac{3}{10}$

Example $\PageIndex{10}$

Solve: $\sqrt{3y+5}+2=5$.

Example $\PageIndex{11}$

Solve: $\sqrt{3p+3}+3=5$.

$\frac{1}{3}$

Example $\PageIndex{12}$

Solve: $\sqrt{5q+1}+4=6$.

$\frac{3}{5}$

When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.

Example $\PageIndex{13}$

Solve: $\sqrt{9k−2}+1=0$.

Example $\PageIndex{14}$

Solve: $\sqrt{2r−3}+5=0$

no solution

Example $\PageIndex{15}$

Solve: $\sqrt{7s−3}+2=0$.

Definition: BINOMIAL SQUARES

\[\begin{array}{cc} {(a+b)^2=a^2+2ab+b^2}&{(a−b)^2=a^2−2ab+b^2}\\ \nonumber \end{array}\]

Don’t forget the middle term!

Example $\PageIndex{16}$

Solve: $\sqrt{p−1}+1=p$.

Example $\PageIndex{17}$

Solve: $\sqrt{x−2}+2=x$.

Example $\PageIndex{18}$

Solve: $\sqrt{y−5}+5=y$.

Example $\PageIndex{19}$

Solve: $\sqrt{r+4}−r+2=0$.

Example $\PageIndex{20}$

Solve: $\sqrt{m+9}−m+3=0$.

Example $\PageIndex{21}$

Solve: $\sqrt{n+1}−n+1=0$

When there is a coefficient in front of the radical, we must square it, too.

Example $\PageIndex{22}$

Solve: $3\sqrt{3x−5}−8=4$.

Example $\PageIndex{23}$

Solve: $\sqrt{24a+2}−16=16$.

$\frac{127}{2}$

Example $\PageIndex{24}$

Solve: $\sqrt{36b+3}−25=50$.

$\frac{311}{3}$

Example $\PageIndex{25}$

Solve: $\sqrt{4z−3}=\sqrt{3z+2}$.

Example $\PageIndex{26}$

Solve: $\sqrt{2x−5}=\sqrt{5x+3}$.

Example $\PageIndex{27}$

Solve: $\sqrt{7y+1}=\sqrt{2y−5}$.

Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again.

Example $\PageIndex{28}$

Solve: $\sqrt{m}+1=\sqrt{m+9}$.

Example $\PageIndex{29}$

Solve: $\sqrt{x}+3=\sqrt{x+5}$.

Example $\PageIndex{30}$

Solve: $\sqrt{m}+5=\sqrt{m+16}$.

Example $\PageIndex{31}$

Solve: $\sqrt{q−2}+3=\sqrt{4q+1}$.

Example $\PageIndex{32}$

Solve: $\sqrt{y−3}+2=\sqrt{4y+2}$.

no solution

Example $\PageIndex{33}$

Solve: $\sqrt{n−4}+5=\sqrt{3n+3}$.

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.

We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.

Definition: SOLVE APPLICATIONS WITH FORMULAS.

We used the formula A=L·W to find the area of a rectangle with length L and width W . A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a square, the area of the square is $ s^2$ .

$This figure shows a square with two sides labeled s. It also indicates that A equals s squared.$

The formula $A=s^2$ gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for s .

\[\begin{array}{ll} {}&{A=s^2}\\ {\text{Take the square root of both sides.}}&{\sqrt{A}=\sqrt{s^2}}\\ {\text{Simplify.}}&{s=\sqrt{A}}\\ \nonumber \end{array}\]

We can use the formula $s=\sqrt{A}$ to find the length of a side of a square for a given area.

Definition: AREA OF A SQUARE

$This figure shows a square with two sides labeled s. The figure also indicates, “Area, A,” “A equals s squared,” “Length of a side, s,” and “s equals the square root of A.”$

We will show an example of this in the next example.

Example $\PageIndex{34}$

Mike and Lychelle want to make a square patio. They have enough concrete to pave an area of 200 square feet. Use the formula $s=\sqrt{A}$ to find the length of each side of the patio. Round your answer to the nearest tenth of a foot.

Example $\PageIndex{35}$

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. Use the formula $s=\sqrt{A}$ to find the length of each side of her lawn. Round your answer to the nearest tenth of a foot.

Example $\PageIndex{36}$

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. Use the formula $s=\sqrt{A}$ to find the length of each side of his mosaic. Round your answer to the nearest tenth of a foot.

Another application of square roots has to do with gravity.

Definition: FALLING OBJECTS

On Earth, if an object is dropped from a height of hh feet, the time in seconds it will take to reach the ground is found by using the formula,

$t=\frac{\sqrt{h}}{4}$

For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting h=64 into the formula.

It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.

Example $\PageIndex{37}$

Christy dropped her sunglasses from a bridge 400 feet above a river. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the sunglasses to reach the river.

Exercise $\PageIndex{38}$

A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the ground.

Example $\PageIndex{39}$

A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula $t=\frac{\sqrt{h}}{4}$ to find how many seconds it took for the squeegee to reach the sidewalk.

3.5 seconds

Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.

Definition: SKID MARKS AND SPEED OF A CAR

If the length of the skid marks is d feet, then the speed, s , of the car before the brakes were applied can be found by using the formula,

$s=\sqrt{24d}$

Example $\PageIndex{40}$

After a car accident, the skid marks for one car measured 190 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Example $\PageIndex{41}$

An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula $s=\sqrt{24d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Example $\PageIndex{42}$

The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula $s=\sqrt{24d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Key Concepts

Check the answer. Some solutions obtained may not work in the original equation.

$This figure shows a square with two sides labeled, “s.” The figure also says, “Area, A,” “A equals s squared,” “Lenth of a side, s,” and “s equals the square root of A.”$

IMAGES

How to Solve Square Root Problems
$solve math problems square roots$
Square Roots
😎 Solving square root problems. Solving Square Roots: Simplification, Addition, Subtraction
Solving Equations with Square Roots (Simplifying Math)
🎉 How to solve a square root problem. Solving Square Roots: Simplification, Addition
Solving by Square Roots Practice

VIDEO

Solving Square Root Equations
Alg 1 Solve by Taking Square Roots
Solving Square Root Equations Level 1
Math 4
Evaluating Some Radical Expression
A Nice Radical Problem • Infinite Square Root

COMMENTS

How to Solve Square Root Problems (with Pictures)
To solve square root problems, understand that you are finding the number that, when multiplied by itself, equals the number in the square root. For quick
Solving square-root equations: one solution (video)
We're asked to solve the equation, 3 plus the principal square root of 5x plus 6 is equal to 12. And so the general strategy to solve this type of equation is
Solving square-root equations (article)
We can use a linear approximation to find a close estimate for the square root. √(x) ≈ (x + y) / (2 * √(y)) where y is a number that is "close to" x
How To Simplify Square Roots
This math video tutorial explains how to simplify square roots.My E-Book: https://amzn.to/3B9c08zVideo Playlists: https://www.video-tutor.
Simplifying square root
A Special Type of Nested Roots and Radicals · Simplify this Cube Root | Step-by-Step Explanation · What Is The Shaded Area? · Math Riddle - Can you
Square Roots
How to simplify problems with square roots and radicals - properties and rules of radicals and square roots.For more in-depth math help
Art of Problem Solving: Square Root Introduction Part 1
Art of Problem Solving's Richard Rusczyk introduces square roots. Love math? Learn more: http://bit.ly/ArtofProblemSolving.
Square Root Calculator
For example, the square roots of 9 are -3 and +3, since (-3)2 = (+3)2 = 9. Any nonnegative real number x has a unique nonnegative square root r; this is called
Square Roots Calculator
Math notebooks have been around for hundreds of years. You write down problems, solutions and notes to go back... Read More.
9.6: Solve Equations with Square Roots
An equation in which the variable is in the radicand of a square root is called a radical equation. As usual, in solving these equations, what