9-2 & 9-3: The Quadratic Formula
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. 149-150: 21-46
• 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
Song 1:
Song 2:
Exercise 1
Write a quadratic equation in standard form
that has the given solutions.
1.
9± 249
14
2.
−3± 361
16
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 2
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
9-2 & 9-3: The Quadratic Formula
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. 149-150: 21-46
• Complex Zeros and De
Moivre WS