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Solving a Quadratic Equation using Square Roots

Author: Sophia
PDF Version

what's covered
This tutorial covers how to solve a quadratic equation using square roots, through the discussion of:

Table of Contents

1. Square Roots: A Review

In review, the square root of x is a number whose product with itself is x.

EXAMPLE

As shown below, 3 times 3 is 9 and -3 times -3 is 9. Therefore, the square root of 9 equals 3 and the square root of 9 also equals -3.

table attributes columnalign left end attributes row cell 3 times 3 equals 9 space space space space space space space space space space left parenthesis short dash 3 right parenthesis left parenthesis short dash 3 right parenthesis equals 9 end cell row cell square root of 9 equals 3 space space space space space space space space space square root of 9 equals short dash 3 end cell end table

did you know
There are always two square roots for any positive number. You can use a plus/minus symbol as a shorthand to indicate the positive and negative solutions, writing the square root of 9, for example, as:

square root of 9 equals plus-or-minus 3

2. Using Square Roots to Solve Quadratic Equations

Although factoring is a good method for solving a quadratic equation, when there’s no middle term or x term, you can solve the quadratic equation by isolating the variable. Isolating the variable uses inverse operations to undo operations applied to the variable. Inverse operations include:

  • Addition/Subtraction
  • Multiplication/Division
  • Squaring/Taking the Square Root.

EXAMPLE

Suppose you want to solve the equation:

2 x squared minus 8 equals 0

You can start to isolate the x variable by adding 8 to both sides of the equation.

table attributes columnalign left end attributes row cell 2 x squared minus 8 equals 0 end cell row cell space space space space space space plus 8 space space space plus 8 end cell row cell 2 x squared equals 8 end cell end table

Next, you can divide both sides by 2, to further isolate the x variable.

table attributes columnalign left end attributes row cell fraction numerator 2 x squared over denominator 2 end fraction equals 8 over 2 end cell row cell x squared equals 4 end cell end table

Finally, you can take the square root of both sides to undo the 2 exponent that is squaring the x. This gives you a solution of x equals a positive or negative 2, so this equation has two solutions.

table attributes columnalign left end attributes row cell square root of x squared end root equals square root of 4 end cell row cell x equals plus-or-minus 2 end cell end table

hint
Remember, taking the square root of a positive number gives two results, one positive and the other negative.

3. Square Root of an Expression

You can also solve a quadratic equation by taking the square root of an expression.

EXAMPLE

Suppose you want to solve the expression:

left parenthesis 3 x plus 3 right parenthesis squared equals 36

Notice that there are multiplication and addition operations being applied to the x variable in the parentheses, which means that you undo them last, because, in general, you use the reverse order of operations when solving an equation.

To solve the equation, start by canceling out the squaring operation by taking the square root of both sides. When you do this, on the left side, you have 3x plus 3, because squaring and square root are inverse operations that cancel each other out. On the right side, you have a positive or negative 6.

table attributes columnalign left end attributes row cell square root of left parenthesis 3 x plus 3 right parenthesis squared end root equals square root of 36 end cell row cell 3 x plus 3 equals plus-or-minus 6 end cell end table

This means that you have two results, and solving each will give provide two solutions. To solve the first equation, start by subtracting 3 on both sides, then dividing by 3 on both sides, which provides x equals 1 for your first solution.

table attributes columnalign left end attributes row cell 3 x plus 3 equals 6 end cell row cell space space space space space minus 3 space space minus 3 end cell row cell space space space space 3 x equals 3 end cell row cell fraction numerator 3 x over denominator 3 end fraction equals 3 over 3 end cell row cell space space space space space x equals 1 end cell end table

To solve the second equation, you also begin by subtracting 3 on both sides, then dividing by 3 on both sides, to give you x equals -3 for your second solution.

table attributes columnalign left end attributes row cell 3 x plus 3 equals short dash 6 end cell row cell space space space space space minus 3 space space minus 3 end cell row cell space space space space 3 x equals short dash 9 end cell row cell fraction numerator 3 x over denominator 3 end fraction equals fraction numerator short dash 9 over denominator 3 end fraction end cell row cell space space space space space x equals short dash 3 end cell end table

You can verify your solution by substituting both solutions separately back into the original equation. Starting with your first solution, substituting x equals 1 into your equation provides:

table attributes columnalign left end attributes row cell left parenthesis 3 left parenthesis 1 right parenthesis plus 3 right parenthesis squared equals 36 end cell row cell left parenthesis 3 plus 3 right parenthesis squared equals 36 end cell row cell 6 squared equals 36 end cell row cell 36 equals 36 end cell end table

Your solution of 1 gives you a true statement, which means it is correct.

Next, check your second solution, x equals -3, by substituting it into the equation.

table attributes columnalign left end attributes row cell left parenthesis 3 left parenthesis negative 3 right parenthesis plus 3 right parenthesis squared equals 36 end cell row cell left parenthesis negative 9 plus 3 right parenthesis squared equals 36 end cell row cell left parenthesis negative 6 right parenthesis squared equals 36 end cell row cell 36 equals 36 end cell end table

Your solution of -3 also gives you a true statement, so both solutions are correct.

summary
Today you reviewed square roots, noting that the square root of x is a number whose product with itself is x. You also learned that when solving quadratic equations with no middle or x term, it is easiest to solve by isolating the variable, using a inverse operation such as taking the square root. Lastly, you learned that taking the square root of an expression when solving an equation may lead to having two solutions to the equation.

Source: This work is adapted from Sophia author Colleen Atakpu.

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