Solving Quadratic Equations by Factoring
Solving an Equation of the Form ax2 + bx + c = 0
by Factoring
The number 0 has several special properties, including this one:
• The product of 0 and any number is 0. For example, 2
· 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 — 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
· 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 —
Quadratic Equation
A quadratic equation is an equation that can be written in the form
ax2 + 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 x2 + 5x -
3x - 15 = 0 x2 + 2x - 15 = 0 |
This quadratic equation is written in standard form.
Notice that the terms on the left side of ax2 + bx + c = 0 are
arranged in descending order by degree. The right side of the equation is zero.
Quadratic equations are also called second-degree equations because
the degree of the polynomial, ax2 + bx + c, is 2.
We will use the Zero Product Property to solve some quadratic equations by
factoring.
Procedure — To Solve a Quadratic Equation By Factoring
Step 1 Write the quadratic equation in the form ax2 + 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.
|
