Download Solve Quadratic Equations Using the Zero

Math 35
5.9 "Solving Equations by Factoring"
Objectives:
* Solve quadratic equations using the zero-factor property.
* Solve higher-degree polynomial equations by factoring.
* Use quadratic equations to solve problems.
Solve Quadratic Equations Using the Zero-Factor Property
De…nition:
"Quadratic Equations"
A quadratic equation is an equation that can be written in the standard form
;
where a; b; and c represent real numbers and a 6= 0:
Examples:
A solution of a quadratic equation is a value of the variable that makes the equation true. Many quadratic equations
can be solved by factoring and using the zero-factor property.
The Zero-Factor Property:
When the product of two real numbers is 0, at least one of them is 0.
:
This property also applies to three or more factors.
The Factoring Method for Solving a Quadratic Equation:
Step 1: Write the equation in standard form: ax2 + bx + c = 0:
Step 2:
Factor the polynomial.
Step 3:
Use the zero-factor property to set each factor equal to zero.
Step 4:
Solve each resulting linear equation.
Step 5:
Check the results in the original equation.
Example 1: (Solving equations by factoring)
Solve the following equations and check the answer.
a) x2 + 5x + 6 = 0
b) x2
4x
45 = 0
Example 2: (Solving equations by factoring)
Solve the following equations and check the answer(s).
a) 4p2 = 12p
b) 16a2 = 49
Page: 1
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.9
Example 3: (Solving equations by factoring)
Solve the following equations and check the answer(s).
6 6
a) x = + x2
5 5
b)
4 2
6
x +x=
5
5
Example 4: (Solving equations by factoring)
Solve the following equation:
2
(2x + 1) =
8 (2x + 1)
16
Example 5: (Solving equations by factoring)
Let f (x) = x2
71: For what value(s) of x is f (x) = 10?
Page: 2
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.9
Solve Higher-Degree Polynomial Equations by Factoring
Some equations involving polynomials with degrees higher than 2 can be solved by factoring. In such cases, we use the
extension of the zero-factor property. When the product of two or more real numbers is 0, at least one of them is 0:
Example 6: (Solving higher-degree polynomial equations)
Solve the following equations.
a) x3
2x2 =
b) 5x3 + 13x2 = 6x
x
Example 7: (Solving higher-degree polynomial equations)
Solve the following equation:
x4 + 5x2
4=0
Example 8: (Solving applied problems)
If the initial velocity of an object launched from the ground straight up into the air is 176 feet per second, when will the
object strike the ground? Hint: Use the equation h =
16t2 + vt
Page: 3
Notes by Bibiana Lopez
Intermediate Algebra by Tussy and Gustafson
5.9
Use Quadratic Equations to Solve Problems
Example 9: (Solving applied problems)
The length of the base of a stained glass widow is 3 times its height, and its area is 96 square feet. Find the length of its
base and its height.
Example 10: (Solving applied problems)
A rectangular shaped spa, 5 feet wide and 6 feet long, is surrounded by decking of uniform width, as shown in the illustration.
If the total area of the deck is 60 ft2; how wide is the decking?
Page: 4
Notes by Bibiana Lopez