Solving Quadratic Equations by Factoring
Solving an Equation of the Form ax^{2} + bx + c = 0
by Factoring
The number 0 has several special properties, including this one:
â€¢ The product of 0 and any number is 0. For example, 2
Â· 0 = 0.
Conversely, if the product of two real numbers is 0, then one or both must
be 0. This statement is known as the Zero Product Property.
Property â€” Zero Product Property
English If the product of two numbers is 0, then one or both of the
numbers is 0.
Algebra If a and b are real numbers, and if a
Â· b = 0, then a = 0 or
b = 0 or both a and b are equal to 0.
Example If (x + 2)(x + 3) = 0, then x + 2 = 0 or x + 3 = 0.
The Zero Product Property is useful for solving certain types of equations.
Example
Solve: (x  3)(x + 5) = 0
Solution
The product of (x  3) and (x + 5) is zero.

(x  3)(x + 5) = 0 
Use the Zero Product Property to write
two separate equations.
Solve each equation. 
x  3 = 0
x = 3 
or
or 
x + 5 = 0 x = 5 
Thus, the original equation has two solutions: 3 and 5.The solutions may be checked by substituting each value of x into the
original equation and simplifying.
Check x = 3. 
Check x = 5. 
(x  3)(x + 5) 
= 0 
(x  3)(x + 5) 
= 0 
Is Is
Is 
(3  3)(3 +5)
(0)(8)
0 
= 0 ? = 0 ?
= 0 ? Yes 
Is Is
Is 
(5  3)(5 + 5)
(8)(0)
0 
= 0 ? = 0 ?
= 0 ? Yes 
Definition â€”
Quadratic Equation
A quadratic equation is an equation that can be written in the form
ax^{2} + bx + c = 0, where a, b, and c are real numbers and a
≠ 0.
A quadratic equation written in this form is said to be in standard
form.
Now we will rewrite this equation.
We multiply the binomials.
We combine like terms. 
(x  3)(x + 5) = 0 x^{2} + 5x 
3x  15 = 0 x^{2} + 2x  15 = 0 
This quadratic equation is written in standard form.
Notice that the terms on the left side of ax^{2} + bx + c = 0 are
arranged in descending order by degree. The right side of the equation is zero.
Quadratic equations are also called seconddegree equations because
the degree of the polynomial, ax^{2} + bx + c, is 2.
We will use the Zero Product Property to solve some quadratic equations by
factoring.
Procedure â€” To Solve a Quadratic Equation By Factoring
Step 1 Write the quadratic equation in the form ax^{2} + bx + c
= 0.
Step 2 Factor the polynomial.
Step 3 Use the Zero Product Property.
Step 4 Solve each equation.
Step 5 Check each answer.
