Section 4.6
The Quadratic Formula
Objectives
 Use the discriminant to determine the
nature of the roots of a quadratic
equation (MA.912.A.7.4)
– Solve quadratic equations using the Quadratic
Formula
– Determine the number of solutions by using the
discriminant
Definition and Review
The quadratic formula is used to find solutions to a quadratic
equation of the form ax2 + bx + c = 0 when
a ≠ 0. The quadratic formula is:
 b  b 2  4ac
x
2a
The expression b2 – 4ac, where a, b, and c are coefficients of
the quadratic equation, is called the __________________.
If b2 - 4ac > 0, the equation has _____ real solution(s).
If b2 - 4ac = 0, the equation has _____ real solution(s).
If b2 - 4ac < 0, the equation has _____ real solution(s).
(It actually has ______________________________.)
Example
Find the discriminant and solve.
1.
x2  4 x  4  0
2. 3x  3x  5
2
Checkpoint
Find the discriminant and solve.
1. 2 x 2  x  x 2  2 x  4
2. x  64  16
2
3.
x  2x  5
2
Using the Discriminant
Graph each of the equations from the
previous slide in your calculator. What
conclusions can you make about the
discriminant and the graph of each
equation?
What about the discriminant and the type of
roots?
Application
The water in a large fountain leaves the spout with
a vertical velocity of 30 feet per second. After
going up in the air it lands in a basin 6 feet below
the spout. If the spout is 10 feet above the
ground, how long does it take a single drop of
water to travel from the spout to the basin? Use
the model h = -16t2 + v0t + h0.
Applications
Use the falling object model h  16t 2  h0
where h is the height (in feet) of the object
after t seconds and h0 is the object’s initial
height.
A person is trapped in a building 120 feet above
the ground and wants to land on a rescue
team’s air cushion. How long before the
person reaches safety?
Application w/ Gravity Function
The tallest building in the United States is in Chicago,
Illinois. It is 1,450 ft. tall.
a) How long would it take a penny to drop from the
top of the building?
b) How fast would the penny be traveling when it
hits the ground if the speed is given by s = 32t
where t is the number of seconds since the penny
was dropped?