4.8: Use the Quadratic Formula and the Discriminant Assignment:

4.8: Use the Quadratic Formula and the
Discriminant
Objectives:
1. To derive and use the
quadratic formula to
solve any quadratic
equation
2. To use the
discriminant to
determine the number
of solutions to a
quadratic equation
Assignment:
• P. 296-299: 1, 2, 3-51
M3, 52-54, 56, 60, 64,
73-75
• Complex Zeros and
De Moivre WS
Warm-Up
In the equation
shown, for what
values of c does
the equation
have 2 real
solutions, 2
imaginary
solutions, or 1
real solution?
x2  8x  c  0
Objective 1
You will be able to derive
and use the quadratic
formula to solve any
quadratic equation
Exercise 1
Solve by completing the square.
3x2 + 8x – 5 = 0
Investigation: A Formula
Solving a quadratic equation by completing
the square is quite useful since it allows
you to solve just about any quadratic
equation. However, it can be cumbersome
and tedious, especially if there are
ungainly fractions involved. What we need
is a formula.
Investigation: A Formula
On your own, try to derive the quadratic
formula. To do this, try completing the
square on the general quadratic equation
in standard form as shown below. Even
though there are variables everywhere, the
technique is still the same as if the a, b,
and c were good old-fashioned numbers.
ax2 + bx + c = 0
Investigation: A Formula
ax2 + bx + c = 0
The Quadratic Formula
Let a, b, and c be real numbers, with a ≠ 0.
The solutions to the quadratic equation
ax2 + bx + c = 0 are
b  b  4ac
x
2a
2
Song 1:
Song 2:
Exercise 1
Solve using the quadratic formula.
x2 – 5x = 7
Exercise 2
Solve using the quadratic formula.
1. x2 = 6x – 4
2. 4x2 – 10x = 2x – 9
3. 7x – 5x2 – 4 = 2x + 3
Exercise 3
Write a quadratic equation in standard form
that has the given solutions.
1.
9± 249
14
2.
−3± 361
16
Launching Stuff
Perhaps a bit more fun than dropping stuff (like
eggs) is launching stuff (also eggs). Here, our
equation must have an initial velocity, v0.
h  16t  v0t  h0
2
Exercise 4
The height in feet of an object projected
vertically upward is given by the equation
g 2
h   t  v0t  h0
2
where g is the acceleration due to gravity (in
feet per second squared), v0 is the object’s
initial velocity (in feet per second), t is the
time in motion (in seconds), and h0 is the
initial height (in feet).
Exercise 4a
1. An astronaut standing on the surface of
Earth’s moon throws a rock vertically into
space. How long will it take the rock to hit
the moon’s surface if the rock is thrown at
an initial velocity of 40 feet per second, at
a height of 5 feet, and the acceleration
due to gravity on the moon is 5.3 feet per
second squared?
Exercise 4b
2. The acceleration due to gravity on Earth is
32 feet per second squared. If the rock
had been thrown on Earth with the same
initial velocity and height, how long will it
take the rock to hit Earth’s surface?
Exercise 4c
3. Compare your answers from parts a and
b. What can be said about the gravity on
Earth compared to the gravity on the
moon?
Exercise 5
Based on the previous Exercise,
1. How can the quadratic formula tell you
how many solutions to expect?
2. How can the quadratic formula tell you
what kind of solutions to expect: Real or
imaginary, rational or irrational?
3. How are the roots related to each other if
they are irrational or imaginary?
Objective 2
You will be able to use the
discriminant to determine the
number of solutions to a
quadratic equation
The Discriminant
Discriminant
In the quadratic formula, the expression
b2 – 4ac is called the discriminant.
Exercise 6
Find the discriminant of the quadratic
equation and give the number and type of
solutions of the equation.
1. x2 + 10x + 23 = 0
2. x2 + 10x + 25 = 0
3. x2 + 10x + 27 = 0
Exercise 7
Find the values of k such that the equation
has a) two real solutions, b) one real
solution, and c) two imaginary solutions.
x2 – 2kx + k = 0
4.8: Use the Quadratic Formula and the
Discriminant
Objectives:
1. To derive and use the
quadratic formula to
solve any quadratic
equation
2. To use the
discriminant to
determine the number
of solutions to a
quadratic equation
Assignment
• P. 296-299: 1, 2,
3-51 M3, 52-54,
56, 60, 64, 73-75
• Complex Zeros
and De Moivre WS
“You’re learning about ants again?”