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2 Tutorials that teach Solving a Quadratic Equation by Factoring
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Solving a Quadratic Equation by Factoring

Solving a Quadratic Equation by Factoring

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In this lesson, students will learn how to solve a quadratic equation by using the factoring method.

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Tutorial
This tutorial covers how to solve a quadratic equation by factoring, through the exploration of:
  1. Solutions to Quadratic Equations
  2. The Zero Product Principle
  3. Solving Quadratic Equations by Factoring


1. Solutions to Quadratic Equations

An equation is a mathematical statement that two quantities have the same value. An equal sign between the two quantities is used to show that they are equal.

Quadratic equations can be written in the following form, in which the variables a, b, and c are real numbers. When quadratic equations are set to 0, the solutions are the values for x that make the expression equal to 0. Therefore, the solutions are commonly referred to as zeroes or roots.

a x squared plus b x plus c equals 0

Quadratic equations are set equal to 0 in order to solve them using different methods, including factoring and the quadratic formula. This lesson focuses on solving quadratic equations by factoring.


2. The Zero Product Principle

Solving quadratic equations by factoring uses the zero product principle.

The zero product principle states that if a product of two factors is zero, then one of the factors must be zero.

This is because anything or any value x multiplied by 0 equals 0. So, zero can be expressed as the product of zero in any other real number x.

table attributes columnalign left end attributes row cell x times 0 equals 0 end cell row cell space space space space 0 equals 0 times x end cell end table
The zero product principle can only be used when one side of the quadratic equation is equal to 0.

Suppose you want to solve the following quadratic equation. The zero product principle tells you that solutions to this equation exist when x plus 7 equals 0 and when x minus 4 equals 0.
table attributes columnalign left end attributes row cell left parenthesis x plus 7 right parenthesis left parenthesis x minus 4 right parenthesis equals 0 end cell row cell x plus 7 equals 0 end cell row cell x minus 4 equals 0 end cell end table

Therefore, you can solve each of these equations to find your two solutions. Solving the first equation, start by subtracting 7 on both sides, which gives you x equals -7 for your first solution. For the second equation, add 4 to both sides, which gives you x equals 4. This means that the quadratic equation has two solutions: x equals -7 and x equals 4.
table attributes columnalign left end attributes row cell x plus 7 equals 0 space space space space space space space space space space x minus 4 equals 0 end cell row cell space space minus 7 space space minus 7 space space space space space space space space space space space plus 4 space space space plus 4 end cell row cell space space space x equals negative 7 space space space space space space space space space space space space space space space space x equals 4 end cell end table


3. Solving Quadratic Equations by Factoring

Suppose you want to solve the following quadratic equation by factoring:

x squared plus 7 x plus 10 equals 0

If you can factor the quadratic expression on the left side of the equation, then you can use the zero product principle to find solutions. Therefore, you want to find two numbers that multiply to 10 and add to 7.

Start by listing the pairs of numbers that multiply to 10, then find the pair of numbers that also adds to 7.

File:1295-solvquad2.PNG

Now you can rewrite your expression in factored form as the following, which will be equal to 0.

left parenthesis x plus 2 right parenthesis left parenthesis x plus 5 right parenthesis equals 0

Next, set each factor equal to 0 to write two separate equations, and solve accordingly, to arrive at your two solutions, x equals -2 and x equals -5.

table attributes columnalign left end attributes row cell x plus 2 equals 0 space space space space space space space space space space x plus 5 equals 0 end cell row cell space space minus 2 space space minus 2 space space space space space space space space space space space minus 5 space space space minus 5 end cell row cell space space space x equals negative 2 space space space space space space space space space space space space space space space x equals negative 5 end cell end table

You can verify each solution by substituting it back into the original equation and simplifying. Substituting your first solution of -2 into the equation provides:

table attributes columnalign left end attributes row cell left parenthesis negative 2 right parenthesis squared plus 7 left parenthesis negative 2 right parenthesis plus 10 equals 0 end cell row cell 4 minus 14 plus 10 equals 0 end cell row cell 0 equals 0 end cell end table

Your final equation is 0 equals 0, which is a true statement, so x equals -2 is correct.

Substituting -5, your second solution, into the equation provides:

table attributes columnalign left end attributes row cell left parenthesis negative 5 right parenthesis squared plus 7 left parenthesis negative 5 right parenthesis plus 10 equals 0 end cell row cell 25 minus 35 plus 10 equals 0 end cell row cell 0 equals 0 end cell end table

Again, 0 equals 0, so your second solution of -5 is also correct.

This next example involves solving an equation that isn’t initially set to equal 0 so requires an additional step in the process.

x squared plus 5 x minus 24 equals 12

You know that to use the zero product principle, you need to set the equation equal to 0. To do this, you can subtract 12 from both sides. Now you can determine if you can factor the quadratic expression to use the zero product principle to find the solutions.

table attributes columnalign left end attributes row cell x squared plus 5 x minus 24 equals 12 end cell row cell space space space space space space space space space space space space minus 12 space minus 12 end cell row cell x squared plus 5 x minus 36 equals 0 end cell end table

Next, you need to find two numbers that multiply to -36 and add to 5. The factor pairs of -36 are listed here, and the pair of numbers that also add to 5 are (-4, 9).

File:1296-solvquad4.PNG

Now you can factor your expression as:

left parenthesis x minus 4 right parenthesis left parenthesis x plus 9 right parenthesis equals 0

Set each factor equal to 0 to write two separate equations, and solve accordingly, to arrive at your two solutions, x equals 4 and x equals -9.

table attributes columnalign left end attributes row cell x minus 4 equals 0 space space space space space space space space space space x plus 9 equals 0 end cell row cell space space plus 4 space space plus 4 space space space space space space space space space space space minus 9 space space space minus 9 end cell row cell space space space x equals 4 space space space space space space space space space space space space space space space x equals negative 9 end cell end table
Try to solve the following equation. You’ll notice that each term has common factors that can be factored out.
4 x squared plus 16 x equals 0
Each term has common factors of 4 and x, which can be factored out. Therefore, your equation becomes:
4 x left parenthesis x plus 4 right parenthesis equals 0
Now you can use the zero product principle to set each factor equal to 0 to find the solutions, x equals 0 and x equals -4.
table attributes columnalign left end attributes row cell fraction numerator 4 x over denominator 4 end fraction equals 0 over 4 space space space space space space space space space x plus 4 equals 0 end cell row cell space space space space space space x equals 0 space space space space space space space space space space space space space x equals negative 4 end cell row cell space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space end cell row blank row blank end table

Today you learned that when quadratic equations are set equal to 0, the solutions are the values for x that make the expression equal to 0. These solutions to quadratic equations are commonly referred to as zeroes or roots. You also learned about the zero product principle, which states that if a product of two factors is equal to 0, then one of the factors must be 0. Lastly, you learned that quadratic equations are set equal to 0 in order to solve them using different methods, including solving quadratic equations by factoring.

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