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CCGPS Geometry
Solving Quadratics
Notes and Practice
Name: ________________________________________________ Date: _______________________
Solving Quadratics by Factoring and Graphing
Let’s start by using our knowledge from Unit 5A and graphing quadratics.
Graph the function y = x2 + 2x – 3 using a table.
3
x

1
0
1
fx
Then, find the x-intercepts, which we will call the
zeroes for the rest of this unit.
x = ___ and x = ___
These same zeroes can be found by factoring the quadratic and using the Zero Product
Property.
The property states that if the product of two factors is zero, then at least one of the
factors must be zero.
For instance, if a x b = 0, then either a or b can be zero to make the product zero.
To see how this applies to the above example, let’s set x2 + 2x – 3 equal to zero and
factor it.
(x + 3) (x -1) = 0


If we plug in -3 into (x + 3) or 1 into (x – 1), then the resulting product is 0.
These are the same two zeroes that were found by graphing the function.
Example 1:
Solve: x2 + 3x – 18 = 0
Practice:
1) x2 + 10x + 16 = 0
Example 2:
Solve: 2x2 + 5x – 12 = 0
2) 6x2 + 7x – 3 = 0
3) 5x2 + 10x = 0
CCGPS Geometry
Solving Quadratics
Notes and Practice
Application 1:
A ball is thrown up with an initial velocity of 32 ft/sec at a height of 240 ft. Use the
equation h(t) = -16t2 + vot + ho to find when the ball hits the ground.
Now, let’s say that we wanted to know when the ball was at certain height instead of
hitting the ground.
For instance, if we wanted to catch the ball on its way back down, we would want to
know where it was at 240 feet again (the start height). Instead of setting the function
equal to 0, we would set it equal to 240 and solve by factoring as shown below.
-16t2 + 32t + 240 = 240
-16t2 + 32t = 0
-16t(t – 2) = 0
So, t = 0 or 2. In this case, 2 is the answer, as t = 0 is when the ball was first thrown.
Application 2:
Bill throws a water balloon from his hotel balcony with an initial velocity of 32 ft/sec at a
height of 128 feet. When will the balloon reach his friend whose balcony is at 80 feet
above the ground?
This same approach can be used in finding the intersection of a quadratic function and
any linear function.
For example, what are the solutions of this system of equations?
y = -3x2 + 4x + 20
y = -2x – 4
We will just sent them equal to each other and solve.
-3x2 + 4x + 20 = -2x – 4
-3x2 + 6x + 24 = 0
-3(x2 – 2x – 8) = 0
-3(x – 4)(x + 2) = 0
(x – 4) = 0 and (x +2) = 0
So, x = -2 or 4
CCGPS Geometry
Solving Quadratics
Notes and Practice
Practice: Solve each of the following by factoring.
1) x2 – 7x + 6 = 0
2) 2x2 – 5x + 2 = 0
3) 4x2 – 16x = 0
4) 3x2 +13x + 4 = 0
5) x2 – 6x = 9
6) 5x2 + 22 = 20x +2
7) The dimensions of a rectangular flower garden are 8 m by 15m. Each dimension was
increased by the same amount. The garden now has an area of 198 m2. Find the
dimensions of the new garden.
8) A rectangular pool measures 5 yd by 6 yd. A concrete deck is constructed around
the pool. The deck and pool together cover an area of 72 yd.2 How wide is the
deck?