Avon High School
Section: 1.5
ACE COLLEGE ALGEBRA II - NOTES
Quadratic Equations
Mr. Record: Room ALC-129
Day 1 of 1
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written in the general from
ax 2  bx  c  0,
where a, b, and c are real numbers, with a ≠ 0. A quadratic equation in x is also called a
second-degree polynomial equation in x.
The Zero-Product Principle
If the product of two algebraic expressions is zero, then at least one of the
factors is equal to zero.
If AB  0, then A  0 or B  0.
Example 1
Solving Quadratic Equations by Factoring
Solve by factoring.
a. 3x 2  9 x  0
b. 2 x 2  x  1
Investigation: Use a graphing calculate to sketch each problem in
Example 1. Where do you see the solutions?
The Square Root Property
If u is an algebraic expression and d is a nonzero real number, then u 2  d has exactly two solutions
If u 2  d , then u  d or u   d .
Equivalently,
If u 2  d , then u   d
Example 2
a. 5 x 2  45  0
Solving Quadratic Equations by the Square Root Property
Solve by the square root property.
b.
 x  5
2
 11
How can we solve a quadratic equation, ax 2  bx  c  0 , if it cannot be factored?
You will have two options.
Completing the Square
2
b
If x  bx is a binomial, then by adding   , which is the square of
2
half the coefficient of x, a perfect square trinomial will result. That is,
2
2
2
b
b 
x 2  bx      x   .
2
2 
Example 3
Solving Quadratic Equations by Completing the Square
Solve by completing the square.
a. x 2  6 x  4  0
b. 2 x 2  3x  4  0
The Quadratic Formula
The solution to a quadratic equation in general form ax 2  bx  c  0 , with a ≠ 0, are
given by the quadratic formula:
b  b2  4ac
x
2a
Watch Power Point “Deriving the Quadratic Equation”
Example 4
So
Solving Quadratic Equations by Using the Quadratic Formula
Solve by using the quadratic formula.
a. 2 x 2  2 x  1  0
b. 3x 2  2 x  4  0
The Discriminant
The quantity under the radical, b 2  4ac , tells us a lot about the solutions to the quadratic equation from which
it comes. The table below outlines the various situations.
Example 5
Blood Pressure and Age
The graphs in the figure to the right illustrate a person’s
normal systolic blood pressure, measured in millimeters of
mercury (mm Hg), depends on his or her age. The formula
P  0.006 A2  0.02 A  120 models a man’s systolic pressure,
P, at age A.
a. Find the age to the nearest year, of a man whose normal systolic
blood pressure is 125 mm Hg.
b. Use the graphs to describe the differences between the normal
systolic blood pressures of men and women as they age.