Up

Solving equations by factoring

Now that we know how to factor, we can apply our knowledge to solving quadratic equations that we were unable to solve before.  We will use factoring and the Zero Product Property to solve these equations easily.  Let’s begin by stating the Zero Product Property:

Definition:  Let a and b be any variables, or algebraic expressions.  Then the following is true:


ab=0   if and only if   a=0   or   b=0

The Zero Product Property is critical in our pursuit of solving quadratic equations.  Now let’s go to an example to see how it works.

Example:  Solve the equation \(3{n^2} + 96 = 36n\) by factoring.

Solution:  First manipulate the equation so that it is in standard form

\(3{n^2} + 96 = 36n\)

\(3{n^2} - 36n + 96 = 0\)

We immediately see a factoring opportunity.  All of the terms in the polynomial have a coefficient that is divisible by 3.  We divide both sides by 3 to obtain:

\({n^2} - 12n + 32 = 0\)

This is a less complicated equation.  And we see that we can factor it yet again.  We factor the left side of the equation and we have

\(\left( {n - 8} \right)\left( {n - 4} \right) = 0\)

Good.  We are close to a solution.  Now we implement the Zero Product Property.  We restate the Zero Product Property using our particular expressions:

\(\left( {n - 8} \right)\left( {n - 4} \right) = 0\) if and only if \(n - 8 = 0\) or \(n - 4 = 0\)

So we have that either \(n - 8 = 0\), in which case \(n = 8\), or \(n - 4 = 0\), in which case \(n = 4\). You can check both solutions to see that they are indeed true.

The solution is \(n = 4\) or \(8\). Now we are able to solve quadratic equations that are factorable.  And it’s all because of the Zero Product Property!

Let’s try another example:

Example:  Solve the equation \(2{x^2} = 18\) by factoring.

Solution:  Again, we will put the equation in standard form, factor the left side, then use the Zero Product Property to find the solutions.  We have

\(2{x^2} = 18\)

\(2{x^2} - 18 = 0\), put equation in standard form

\({x^2} - 9 = 0\), divide both sides by 2

\(\left( {x + 3} \right)\left( {x - 3} \right) = 0\), factor the left side

\(x =  - 3or3\), Zero product Property

Bellow you can download some free math worksheets and practice.


Downloads:
4785 x

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-factoring-easy

Watch below how to solve this example:

 

Downloads:
5609 x

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-factoring-medium

Watch below how to solve this example:

 

Downloads:
3417 x

Solve each equation by factoring.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Quadratic-Functions-Solving-equations-by-factoring-hard

Watch below how to solve this example:

 
 
 

Facebook PageYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems