4.8: Use the Quadratic Formula and the Discriminant Objectives: Assignment:

4.8: Use the Quadratic Formula and the
Discriminant
1.
2.
Objectives:
To derive and use
the quadratic
formula to solve any
quadratic equation
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
x
b 
b 2  4ac
2a
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 2
Solve using the quadratic formula.
1.
x2 = 6x – 4
Exercise 2
Solve using the quadratic formula.
2. 4x2 – 10x = 2x – 9
Exercise 2
Solve using the quadratic formula.
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 2  v0t  h0
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 4a
1. v0 = 40 feet per second;
h0 = 5 feet;
gm = 5.3 feet per second2
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 4b
2. v0 = 40 feet per second;
h0 = 5 feet;
gE = 32 feet per second2
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?
Objective 2
You will be able to use the
discriminant to determine the
number of solutions to a
quadratic equation
Exercise 5
Based on the previous Exercises,
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?
Exercise 5
Based on the previous Exercises,
1. How can the quadratic formula tell you
how many solutions to expect?
Exercise 5
Based on the previous Exercise,
2. How can the quadratic formula tell you
what kind of solutions to expect: Real or
imaginary, rational or irrational?
Exercise 5
Based on the previous Exercise,
3. How are the roots related to each other if
they are irrational or imaginary?
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
1.
2.
Objectives:
To derive and use
the quadratic
formula to solve any
quadratic equation
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?”