LESSON 3 - 5 : Solving Quadratic Equations
MCR3U1
MINDS ON...
Mrs. A. throws a football to Ferb. The height of the ball is modelled by the function
h(t) = -5t2 + 20t + 2, with h in metres and t in seconds.
Unfortunately, Ferb would rather hang on to his math textbook than catch the ball.
How long was the ball in the air before it hits the ground?
Usually, we let h = 0 and solve for t:
h(t) = -5t2 + 20t + 2
0 = -5t2 + 20t + 2
How do we solve for t in this case?
When all else fails use the Quadratic Formula!
LESSON 3-5 : Solving Quadratic Equations
MCR3U1

Recall:
y = 5x2 + 40x + 100 is a Quadratic Function (a relation between x and y)
0 = 5x2 + 40x + 100 is a Quadratic Equation (one variable to solve for, x)

Solving a quadratic equation means finding the “zeros” or x-intercepts of the function when y=0.
These zeros are called the solutions or “roots” of the equation.

Quadratic Equations may be solved using various methods:
- Inverse operations, Graphing, Factoring, Completing the Square, and the Quadratic Formula
1. Solving Quadratics by Inverse Operations:
(x – h)2 = c
or ax2 + c = 0
Ex. 1: Solve (x – 1) 2 = 16
2. Solving Quadratics by Graphing
 This method is often time consuming and not accurate
 Example: Solve 0 = x2 –2x – 3 by graphing.
x = -1 and x = 3
3. Solving Quadratics by Factoring
 Solving by factoring is by far the easiest method if the trinomial factors nicely.
 Factor and solve for x using the Zero Product Property: If A x B= 0, then A = 0, B = 0
or both
Ex. 2: Solve by factoring:
a) 0 = x2 + 7x
b) 6x2 + x – 2 = 0
c) 9x2 – 16 = 0
4. Solving Quadratics by Completing the Square
 Complete the square of your quadratic and isolate the variable.
 Remember to keep your solutions in simplest rational or radical form.
Ex. 4: Solve by completing the square.
2x2 - 12x - 14 = 0
Note: If a quadratic gives the roots
and
x = 1.92
and
,
then:
are the EXACT ROOTS
x = -0.42
and
are the APPROXIMATE ROOTS.
5. Solving Quadratics using the Quadratic Formula


 b  b2  4ac
2a
Remember to keep your solutions in simplest rational or radical form.
The formula is:
x 
Ex. 5: Solve by The Quadratic Formula (First, place in standard form to get a, b and c)
a) -x2 - 37 = -14x
Homefun:
b) n2 – 3n = -3
Read p. 172 Ex. 1 and 3. (Note: Break-Even point happens when Profit = 0).
p. 177 #1ad, 2ab, 6b, 7 – 10, 14
EXIT TICKET…
Back to the problem….
Jonathan throws a football to Chanelle. The height of the ball is modelled by the
function h(t) = -5t2 + 20t + 2, with h in metres and t in seconds.
Unfortunately, Chanelle would rather hang on to her math textbook than catch the
ball.
How long does Chanelle have to catch the ball? i.e., How long is the ball in the air
before it hits the ground?
Usually, we let h = 0 and solve for t:
h(t) = -5t2 + 20t + 2
0 = -5t2 + 20t + 2
How do we solve for t in this case?
When all else fails use the Quadratic Formula!
Solve 0 = -5t2 + 20t + 2 :
a = -5
b = 20
c=2
Answers may be stated in exact rational or radical form or approximate decimal form.
Or