4.3 and 4.4: Solving Quadratic Equations

4.3 and 4.4: Solving Quadratic Equations
Objectives:
1. To solve a quadratic
equation by factoring
Assignment:
• P. 255-258: 1, 24-60
M3, 62, 63, 64, 72-74
• P. 263-265: 33-60 M3,
68-70
• Challenge Problems
Warm-Up
If the product of 𝐴 and 𝐵 equals zero, what
must be true about 𝐴 or 𝐵?
Zero Product Property
𝐴∙𝐵 =0
If the product of two
expressions is zero,
then at least of the
expressions equal
zero.
Maybe that’s zero
Or maybe this one’s zero
(Or maybe they’re both zero)
Vocabulary
𝑦-intercepts
𝑥-intercepts
Parabola
Roots (of a function)
Zeros (of a function)
Objective 1
You will be able to solve a
quadratic equation by
factoring
Exercise 1
Use your calculator
to graph the
equation
y = x2 – x – 6.
Where does the
parabola cross the
x-axis?
4
2
-5
5
-2
x-intercepts =
points where
parabola
crosses x-axis
-4
-6
Parabola
The graph of a
quadratic function
is a parabola.
x-intercepts:
where the
parabola
intersects the
x-axis
4
2
-5
5
-2
x-intercepts =
points where
parabola
crosses x-axis
-4
-6
Exercise 2
How many y-intercepts can the graph of the
quadratic function y = ax2 + bx + c have?
– Only one since it’s a function!
How many x-intercepts can the graph of the
quadratic function y = ax2 + bx + c have?
– Two, one, or none!
Exercise 3
If you wanted to find the
y-intercept of a
quadratic function,
what would you do?
Plug in zero for x
and solve for y
Exercise 4
Find the y-intercept of y = x2 – 6x – 7.
Exercise 5
If you wanted to find
the x-intercept of a
quadratic function,
what would you
do?
Plug in zero for y
and solve for x
Exercise 6
Find the x-intercept(s) of y = x2 – 6x – 7.
The problem here is, how do you solve
0 = x2 – 6x – 7 since you can’t just get x by
itself?
The answer is to use the Zero
Product Property
Solving Quadratic Equations
The standard form of a quadratic equation
in one variable is ax2 + bx + c = 0, where a
is not zero.
Solving a quadratic equation in standard
form is the same thing as finding the
x-intercepts of y = ax2 + bx + c.
Solving Quadratic Equations
The standard form of a quadratic equation
in one variable is ax2 + bx + c = 0, where a
is not zero.
We can use the zero product property to
solve certain quadratic equations in
standard form if we can write ax2 + bx + c
as a product of two expressions. To do
that, we have to factor!
Exercise 7
Find the x-intercept(s) of y = x2 – 6x – 7.
Let y = 0
0  x2  6 x  7
0   x  7  x  1
Set each
factor equal
to zero
x7  0
x7
Factor
x 1  0
x  1
x-intercepts: (7, 0) and (−1, 0)
Solving Quadratic Equations
To solve a quadratic
equation, try applying the
zero product property.
Factor
your1
Step
quadratic
Set each
factor
equal to
Step
2
zero and
solve
Exercise 8
Solve 0 = x2 – x – 42.
Same Thing as…
In solving a quadratic
function, you must
find the x-values
that make the
function equal to
zero.
Same as the
roots of the
quadratic
equation
Same as
finding the 𝑥values of the
𝑥-intercepts
Solving
Quadratics
Same as the
zeroes of the
quadratic
function
How Many Solutions?
There can be 2,
1, or 0
solutions to a
quadratic
equation,
depending
upon where it
is in the
coordinate
plane.
Exercise 9
Find the roots of each equation.
1. x2 + 2x = 0
2.
x2 – 12x +36 = 0
3.
x2 + 1 = 0
4.
x2 + 2x + 4 = 0
Exercise 10
Explain why you cannot use the zero
product property to solve every quadratic
equation.
Exercise 11
Find the zeros of each function.
1. y = 16x2 – 4
2.
y = 9x2 + 12x + 4
3.
y = 5x2 + 16x + 3
4.
y = 2x2 + x + 3
Exercise 12
Solve the equation.
1. 4x2 – 17x – 15 = 0
2. 3x2 + 22x + 60 = -14x – 48
Example 13
Find the value(s) of x.
Exercise 14
Find the value of x if the area of the triangle
is 42 square units.
Exercise 15
Use substitution to solve the system of
equations.
4.3 and 4.4: Solving Quadratic Equations
Objectives:
1. To solve a quadratic
equation by factoring
Assignment
• P. 255-258: 1, 2460 M3, 62, 63, 64,
72-74
• P. 263-265: 33-60
M3, 68-70
• Challenge
Problems
“Set it equal to zero!”