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Solving Quadratics by Factoring
There are several ways to solve a quadratic equation. The method explored in this lesson is factoring. The type of factoring chosen to solve a quadratic equation depends upon the equation itself. Regardless of the factoring method chosen, you must start with the equation in standard form. Once it is in standard form, you can factor and use the factored form to solve the equation.

Example
Example
#1: Solve

  • First, the quadratic must be written in standard form .

  • So begin by adding 12 to each side of the equation to obtain
  • When factoring this type of trinomial, the signs in the two sets of parentheses must be alike since the sign of 12 is positive, and both of the signs must be negative since 7x is negative. Also, 4 and 3 must be the factors chosen for 12 since 4 and 3 add to give the 7 in the middle term. If you need additional help with factoring, click here for a factoring lesson.

  • Now that the left side of the equation has been factored, we can use the Zero Product Property. This property says that if a product of two factors is equal to zero, then one or both of the factors must be zero. So we can use this property and set each factor equal to zero. Then solve each equation for x.
What is your answer?
 

You might be wondering why when using the Zero Product Property we use the word “or” and when discussing the solutions we use the word “and”.

The Zero Product Property says that one or both factors can be equal to zero which is why we use the word “or”. This is simply saying that either one of those numbers will make the equation equal to zero.

When solving a quadratic equation, we are looking for any and all values that will make the equation true. That’s why we use “and” when talking about the solutions. Both of those values will make the equation true.

Just remember that when you are stating the solutions for the equation, you should use both values obtained from your factoring.

Examples
Example
#2: Solve
  • Make sure the equation is in standard form. It is, so proceed.
  • Set each factor equal to 0 and solve for x.
What is your answer?
 
Example
#3: Solve
  • Make sure the equation is in standard form. It is, so proceed.
  • Factor the left side using the difference of two squares.
  • Set each factor equal to 0 and solve for x.
What is your answer?
 



S Taylor

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