Solving Quadratic Equations by Factoring – Examples for Your Exam
You will need to study solving quadratic equations by factoring with examples in order to learn this skill for your math exam.
If you do not know how to factor, please see our post on factoring quadratics before continuing.
To see the examples, please go to the sections below the quiz.
Solving Quadratics Equations by Factoring – Quiz
Instructions: Solve the quadratic equations provided below. Then check your answers after each question.
[WpProQuiz 29]
Solving Quadratic Equations by Factoring – Example
If the equation is not equal to zero, you will need to go about solving quadratic equations by factoring using the steps below.
Look at the following example.
Solve for a: (a + 4)(a – 2) = 7
STEP 1: Perform the FOIL Mehtod
Perform the FOIL method on the left-hand side of the equation.
(a + 4)(a – 2) = 7
(a × a) + (a × –2) + (4 × a) + (4 × –2) = 7
a2 – 2a + 4a – 8 = 7
a2 + 2a – 8 = 7
STEP 2: Perform the Operations
Perform addition or subtraction to make the right-hand side of the equation zero.
a2 + 2a – 8 = 7
a2 + 2a – 8 – 7 = 7 – 7
a2 + 2a – 15 = 0
STEP 3: Factor the New Equation
a2 + 2a – 15 = 0
Look at the third term of the equation, which is –15 in our problem.
Factor the Constant
We need to find two integers that equal –15 when they are multiplied.
–1 × 15 = –15
–3 × 5 = –15
–5 × 3 = –15
–15 × 1 = –15
Factor the Coefficient
Then look at the second term of the equation.
We need to be sure that our two integers also equal 2 when they are added.
–1 + 15 = 14
–3 + 5 = 2
–5 + 3 = –2
–15 + 1 = –14
Set up the Factored Form
Finally, take the two integers that meet both conditions above, and set out the factored form of the quadratic equation.
(a – 3)(a + 5) = 0
STEP 4: Solve for the First Set of Parentheses
Solve for a in the left-hand set of parentheses in the newly factored equation.
(a – 3)(a + 5) = 0
(3 – 3) × (a + 5) = 0
(0) × (a + 5) = 0
So, 3 is a solution.
STEP 5: Solve for the Second Set of Parentheses
Solve for a in the right-hand set of parentheses in the newly factored equation..
(a – 3)(a + 5) = 0
(a – 3) × (–5 + 5) = 0
(a – 3) × (0) = 0
So, –5 is another solution.
Solving Quadratic Equations – “Equal to Zero” Method
This method is for quadratic equations in their factored form.
Look at the following example.
Solve for x: (x + 1)(x – 2) = 0
When solving quadratic equations, you will usually need to find two solutions for the squared variable, which is x in our problem.
Remember to solve for zero for each set of parentheses!
You will also need to be sure that the equation equals zero.
When the number to the right of the equals sign is zero, we know that one of the multipliers also needs to be zero.
STEP 1:
Solve for x in the left-hand set of parentheses.
(x + 1)(x – 2) = 0
(–1 + 1) × (x – 2) = 0
(0) × (x – 2) = 0
So, –1 is a solution.
STEP 2:
Solve for x in the right-hand set of parentheses.
(x + 1)(x – 2) = 0
(x + 1) × (2 – 2) = 0
(x + 1) × (0) = 0
So, 2 is also a solution.
Solving Quadratic Equations – Other Methods
To learn how to solve equations in the standard quadratic form, please see our posts on completing the square and the quadratic formula.
You may also want to view our posts on the quadratic equation and algebra.
