Chapter 8
8-1 Introduction to Quadratic Equations
Quadratic Equation
Any equation of degree 2
Definition
An equation of the form ax2 + bx + c = 0, where a, b, and c
are constants and a 0 is called the standard form of a
quadratic equation
Examples:
Objective 1: Solve equations of the type ax2 + bx + c = 0 by factoring
Objective 1: Solve equations of the type ax2 + bx + c = 0 by factoring
Objective 1: Solve equations of the type ax2 + bx + c = 0 by factoring
Objective 2: Solve quadratic equations of the form Ax2 = C by using
square roots
Objective 2: Solve quadratic equations of the form Ax2 = C by using
square roots
Objective 2: Solve quadratic equations of the form Ax2 = C by using
square roots
Objective 2: Solve quadratic equations of the form Ax2 = C by using
square roots
Objective 2: Solve quadratic equations of the form Ax2 = C by using
square roots
Completing the Square: A process by which you force a trinomial
to be a perfect square trinomial so you can solve it using square roots.
Objective 3: Solve quadratic equations by completing the square
Objective 3: Solve quadratic equations by completing the square
Objective 3: Solve quadratic equations by completing the square
Objective 3: Solve quadratic equations by completing the square
HW #8.1
Pg 345-346 3-54 every third
problem, 57-62
Chapter 8
8-2 Using Quadratic Equations
Two cyclists A and B leave the same point one traveling north and the
other traveling west. B travels 7 km/h faster than A. After 3 hours they
are 39 km apart. Find the speed of each cyclist.
Two cyclists A and B leave the same point one traveling north and the
other traveling west. B travels 7 km/h faster than A. After 3 hours they
are 39 km apart. Find the speed of each cyclist.
Scott wants to swim across a river that is 400 meters wide. He begins
swimming perpendicular to the shore he started from but ends up 100
meters down river from where he started because of the current. How far
did he actually swim from his starting point?
In construction, floor space must be given for staircases. If the second
floor is 3.6 meters above the first floor and a contractor is using the
standard step pattern of 28 cm of tread for 18 cm of rise then how many
steps are needed to get from the first to the second floor and how much
linear distance will need to be used for the staircase?
Chapter 8
8-3 Quadratic Formula
Three objects are launched from the top of a 100-foot building.
The first object is launched upward with an initial velocity of 10
feet per second. The second object is dropped. The third object is
launched downward with an initial velocity of 10 feet per second.
HW #8.2-3
Pg 349 8-13
Pg 352-353 3-36 every third, 37-52
Chapter 8
8-4 Solutions of quadratic Equations
Complex conjugate solutions
Determine the nature of the solutions
Theorem 8-4: For the equation ax2 + bx + c = 0, the sum of the
solutions is  b , and the product of the solutions is c .
a
a
Find the sum and the product of the solutions of the following
quadratic equations:
Theorem 8-4: For the equation ax2 + bx + c = 0, the sum of the
solutions is  b , and the product of the solutions is c .
a
a
Find a quadratic equation for which the sum and product of
the solutions is given:
1
1
Sum  - ; Product 
Sum  -5; Product 
4
2
Use the sum and product properties to write a quadratic
equation whose solutions are given.
Use the sum and product properties to write a quadratic
equation whose solutions are given.
Use the sum and product properties to write a quadratic
equation whose solutions are given.
HW #8.4
Pg 357 Left Column, 56-65, 67-72
Chapter 8
8.5 Equations Reducible to Quadratic Form
An equation is said to be in quadratic form if it is reducible to a
quadratic equation through a substitution
x4  9 x2  8  0
Let u =
x2
x3 x 4  0
Let u  x
x 6  6 x3  9  0
Let u  x3
( x 2  1) 2  ( x 2  1)  2  0
Let u  x 2  1
2
5
1
5
x x 2 0
Let u  x
7
1
5
z2  4 7 z  4  0
Let u  7 z
( x  2) 2  2 x  4  1
3
2x  3  6 2x  3
x3 4 x3  2
HW #8.5
Pg 361 1-19 Odd, 20-26
Chapter 8
8.6 Formulas and Problem Solving
Solve for the indicated variable:
V   r 2 h; r
2 r 2  2 rh  1; r
Three objects are launched from the top of a 100-foot building.
The first object is launched upward with an initial velocity of 10
feet per second. The second object is dropped. The third object is
launched downward with an initial velocity of 10 feet per second.
How long will it take each object to hit the ground?
A ladder 10 ft. long leans against a wall. The bottom of the ladder is
6 feet from the wall. How much would the lower end of the ladder
have to be pulled away so that the top end be pulled down by 3 feet?
HW #8.6
Pg 364-365 1-29 Odd, 30
Chapter 8
8.7 Quadratic Variations and
Applications
Definition
Direct Quadratic Variation
Y varies directly as the square of x if there is some
nonzero number k such that y = kx2
Find an equation of variation where y varies directly as the
square of x, and y = 12 when x = 2.
Definition
Direct Quadratic Variation
Y varies directly as the square of x if there is some
nonzero number k such that y = kx2
Find an equation of variation where y varies directly as the
square of x, and y = 175 when x = 5.
Definition
Inverse Quadratic Variation
Y varies inversely as the square of x if there is some
nonzero number k such that y = k/x2
Find an equation of variation where w varies inversely as the
square of d, and W = 3 when d = 5.
Definition
Inverse Quadratic Variation
Y varies inversely as the square of x if there is some
nonzero number k such that y = k/x2
Find an equation of variation where y varies inversely as the
square of x, and y = ¼ when x = 6.
Definition
Joint Variation
Y varies jointly as x and z if there is some nonzero
number k such that y = kxz
Find an equation of variation where y varies jointly as x and z
and y = 42 when x = 2 and z = 3.
Definition
Joint Variation
Y varies jointly as x and z if there is some nonzero
number k such that y = kxz
Find an equation of variation where y varies jointly as x and z
and y = 65 when x = 10 and z = 13.
HW #8.7
Pg 370-371 1-23 Odd
Test Review
Find b and c if the equation 4 x 2  bx  c  0 has
solutions
x
1
7
x

or
2
2
If the sum and the product of the solutions of a quadratic equation are the
same, and one of the solutions is 5, what is the equation?
Write a quadratic equation in standard form that has one solution, 3  i
What is the sum of the reciprocals of the solutions of the equation
2003
1
x 1  0
2004
x
Write a quadratic equation in standard form with integer
coefficients that has two solutions, on of which is
3 i

2 2
HW #R-8
Pg 376 1-30