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Solving Quadratic Equations

Author: Sophia
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what's covered
In this lesson, you will learn how to solve a quadratic equation using the Zero Factor Property of Multiplication. Specifically, this lesson will cover:

Table of Contents

1. Solutions to Quadratic Equations

There are several ways we can talk about solutions to quadratic equations. Solutions are also called roots or zeros because they represent x-values to the equation that makes y equal to zero. Below is a graphical representation of solutions to a quadratic equation:

hint
Quadratic equations will have either one, two, or zero real solutions. We can relate this to the graphical interpretation of solutions. A parabola will intersect the x-axis no more than two times. If a parabola never intersects the x-axis, it has no real solutions.

Two Solutions One Solution No Solutions
Solutions: (-2, 0) and (5, 0) Solutions: (-3, 0) No Real Solutions

2. Zero Factor Property of Multiplication

If any factor of an algebraic expression equals zero, the entire expression has a value of zero. This is because any quantity multiplied by zero is zero. This fact comes in handy when solving quadratic equations written in standard form. We can simply set each factor equal to zero, and solve simpler equations to find solutions to the quadratic.

EXAMPLE

Find the solutions for the quadratic equation x squared minus 4 x minus 21 equals 0.

x squared minus 4 x minus 21 equals 0 Factor the left side of the equation by finding two numbers that multiply to -21 and add to -4
open parentheses x plus 3 close parentheses open parentheses x minus 7 close parentheses equals 0 Set each factor equal to zero
x plus 3 equals 0 comma space space x minus 7 equals 0 Solve for x in each factor
x equals short dash 3 comma space space x equals 7 Our solutions

By setting each factor equal to zero and solving for x, we have found two solutions to the quadratic. When x equals short dash 3 comma the quadratic will equal zero:

table attributes columnalign left end attributes row cell open parentheses x plus 3 close parentheses open parentheses x minus 7 close parentheses equals 0 end cell row cell open parentheses short dash 3 plus 3 close parentheses open parentheses short dash 3 minus 7 close parentheses equals 0 end cell row cell open parentheses 0 close parentheses open parentheses short dash 10 close parentheses equals 0 end cell row cell 0 equals 0 end cell end table

Similarly, when x equals 7 comma the quadratic will equal zero:

table attributes columnalign left end attributes row cell open parentheses x plus 3 close parentheses open parentheses x minus 7 close parentheses equals 0 end cell row cell open parentheses 7 plus 3 close parentheses open parentheses 7 minus 7 close parentheses equals 0 end cell row cell open parentheses 10 close parentheses open parentheses 0 close parentheses equals 0 end cell row cell 0 equals 0 end cell end table

big idea
Use the Zero Factor Property of Multiplication to solve quadratic equations by setting each factor equal to zero. Solving for x when each factor is set equal to zero will reveal solutions to the quadratic.

3. Solving a Quadratic with No Constant Term

When a quadratic is set equal to zero, but has no constant term, making use of the Zero Factor Property is still helpful, but there is some work that needs to be done first. Without the constant term, we know that all of the terms in the quadratic share a factor of x. This means we can factor it out, and then apply the Zero Factor Property.

EXAMPLE

Find the solutions for the quadratic equation 2 x squared minus 6 x equals 0.
2 x squared minus 6 x equals 0 Factor out x
x open parentheses 2 x minus 6 close parentheses equals 0 Set each factor equal to 0
x equals 0 comma space space 2 x minus 6 equals 0 Solve for x in each factor
x equals 0 comma space space 2 x equals 6 Further simplify the second factor
x equals 0 comma space space x equals 3 Our solutions

Again, we can check our solutions by plugging these values back into the quadratic equation.

When x equals 0:

table attributes columnalign left end attributes row cell table attributes columnalign left end attributes row cell 2 x squared minus 6 x equals 0 end cell row cell 2 open parentheses 0 close parentheses squared minus 6 open parentheses 0 close parentheses equals 0 end cell row cell 2 open parentheses 0 close parentheses minus 0 equals 0 end cell row cell 0 minus 0 equals 0 end cell end table end cell row cell 0 equals 0 end cell end table

When x equals 3:

table attributes columnalign left end attributes row cell table attributes columnalign left end attributes row cell 2 x squared minus 6 x equals 0 end cell row cell 2 open parentheses 3 close parentheses squared minus 6 open parentheses 3 close parentheses equals 0 end cell row cell 2 open parentheses 9 close parentheses minus 18 equals 0 end cell row cell 18 minus 18 equals 0 end cell end table end cell row cell 0 equals 0 end cell end table

4. Solving a Quadratic with No x-Term

If a quadratic equation is missing an x-term, using the Zero Factor Property is not as helpful. Instead, we can solve using standard algebraic techniques for solving any equation. The important thing to remember here is that when we apply the square root to a quantity squared, we must include both positive and negative values of the root because a negative value squared is also positive.

EXAMPLE

Find solutions to the quadratic equation x squared minus 7 equals 0.

x squared minus 7 equals 0 Move the constant to the right side by adding 7 to both sides
x squared equals 7 Apply square root to both sides
square root of x squared end root equals square root of 7 Evaluate
x equals plus-or-minus square root of 7 Don't forget to include both positive and negative solutions
x equals short dash square root of 7 comma space space x equals square root of 7 Our solutions

hint
Be sure to include plus or minus when taking the square root in equations. In our example above, both negative square root of 7 and square root of 7 evaluate to 7 when squared.

summary
To find the solutions to quadratic equations algebraically, let y equal 0 and solve the equation for x. The solutions to a quadratic equation are also called the roots or zeros. On a graph, the solutions are the x-intercepts of the parabola. The zero factor property of multiplication says that if any factor is equal to zero, then the entire expression is equal to zero. This property is used when solving a quadratic equation by factoring and can be used when solving a quadratic with no constant term and solving a quadratic with no x-term.

Source: ADAPTED FROM "BEGINNING AND INTERMEDIATE ALGEBRA" BY TYLER WALLACE, AN OPEN SOURCE TEXTBOOK AVAILABLE AT www.wallace.ccfaculty.org/book/book.html. License: Creative Commons Attribution 3.0 Unported License

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