7-3 and 7-4: Solve Quadratic Equations and Inequalities

7-3 and 7-4: Solve Quadratic Equations and
Inequalities
Objectives:
1. To solve a quadratic
equation by factoring
2. To solve quadratic
inequalities
Assignment:
• P. 120: 16-27
• 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)
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!
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 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
Objective 2
You will be able to solve
quadratic inequalities
Less ThAND
GreatOR
Quadratic Inequalities (One)
Solving quadratic inequalities in one variable
is similar to solving a combination of linear
inequalities and absolute value
inequalities.
• Like linear: graphed on a number line
• Like absolute value: involves “and” or “or”
intervals
– “And” = segment; “Or” = 2 rays in opposite
directions
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now taking the square root of both sides
doesn’t translate well with inequalities, so
solve the corresponding quadratic
equation:
x 2  25
x  5
These are our critical values. Graph them.
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now you have 3 intervals to consider.
Which one(s) make(s) the inequality true?
x  5 or x  5
GreatOR
Quadratic Inequalities (One)
Consider the simple quadratic inequality:
x 2  25
Now you have 3 intervals to consider.
Which one(s) make(s) the inequality true?
5  x  5
Less ThAND
Quadratic Inequalities (One)
Another way to think about solving a
quadratic inequality in one variable is to
relate it to a quadratic inequality in two
variables.
Instead of 𝑥 2 − 4 < 0, consider 𝑥 2 − 4 < 𝑦.
Now graph this inequality, shading the
appropriate region.
Quadratic Inequalities (One)
x2  4  0
x2  4  y
The answer is where
the shading touches
the x-axis:
2 x  2
Quadratic Inequalities (One)
x2  4  0
x2  4  y
The answer is where
the shading touches
the x-axis:
x  2 or x  2
Exercise 9
Solve the given inequality.
2x2  7 x  4
Exercise 10
Solve the given inequality.
2x2  2x  3
Exercise 11
Find the domain of each function.
1. 𝑦 = 𝑥 2 − 121
2. 𝑦 = 121 − 𝑥 2
Exercise 12
Find the value(s) of x.
Exercise 13
Use substitution to solve the system of
equations.
7-3 and 7-4: Solve Quadratic Equations and
Inequalities
Objectives:
1. To solve a quadratic
equation by factoring
2. To solve quadratic
inequalities
Assignment:
• P. 120: 16-27
• Challenge Problems