3.2 Quadratic Equations and Problem Solving
Quadratic function: f(x) = ax2 + bx + c
Quadratic equation: ax2 + bx + c = 0 (you solve this
equation to find the intercepts of the quadratic function)
 3x2 + 4x = x2 + 1 is a quadratic equation
 move everything to the LHS, 0 on RHS
 2x2 + 4x - 1 = 0
( a = 2, b = 4, c = -1)
Styles of quadratic equations and their solutions:
1. constant term missing:
3x2 - x = 0
x(3x -1) = 0
factor
x = 0, x = 1/3 set each factor = 0, solve
2. linear term missing:
3x2 - 1 = 0
3x2 = 1
x2 = 1/3 (solve for x2)
1
3
x= 
(simplest radical form)

3
3
3. all terms present, factors easily:
3x2 + 5x - 2 = 0
(3x - 1)(x + 2) = 0
LIOF (reverse FOIL)
x = 1/3, -2
set each factor = 0, solve
4. all terms present, does not factor easily
.002x2 + 15x + 2.56 = 0 maybe from an application
use the quadratic formula
Note: omit Completing the Square (p. 193)
3.2-1
Problem solving: an application
The position equation (or formula) for projectile motion:
A formula that gives the height of an object that has been dropped or thrown
(vertically), at any time after it is released (dropped or thrown).
Symbols in the formula:
t
= time after release (in seconds)
s(t) = height above ground (in feet) after time t
v0 =
initial velocity (positive if upward;
negative if downward, in feet per second)
h0 = initial height above ground (in feet)
The formula (thanks to Sir Isaac Newton):
s(t) = -16t2 + v0t + h0
Example:
George Washington (6 feet tall) throws a silver dollar straight
up into the air with an initial velocity of 46 feet per second. How long after
he throws it will the coin strike the ground?
What we know and don’t know:
symbol value
explanation
t
not known time to strike ground after release is
what we are trying to determine
s(t)
0
the height above ground when it hits
the ground!
v0
46
upward, so positive
h0
6
George is 6 feet tall, so that’s approx.
the height of the coin at release
Substitute in equation:
0 = -16t2 + 46t + 6
Solve: -2(8t2 - 23t - 3) = 0
-2(t - 3)(8t + 1) = 0
t=3
Answer: the dollar will strike the ground after 3 seconds
3.2-2