1. Chapter 4 Quadratic Equations
4.5
The
Quadratic
Formula

2. The solution of the quadratic equation
The Quadratic Formula
The solution of the quadratic equation
ax2 + bx + c = 0 can be found by using the
quadratic formula:
4.5.2

3. Solving Quadratic Equations Using the Quadratic Formula
Solve 2x2 - 5x + 2 = 0.
a = 2, b = -5, c = 2 or
x = 2
4.5.3

4. Solving Quadratic Equations Using the Quadratic Formula
Solve x2 - 6x + 7 = 0.
4.5.4

5. Solving Quadratic Equations With No Real Roots
Solve x2 - 5x + 7 = 0.
Since
is not defined by
real numbers, then
this equation has
NO REAL ROOTS.
4.5.5

6. Practice Time: Worksheet

7. The
Discriminant

8. will give the roots of the quadratic equation.
The Nature of the Roots
The quadratic formula
will give the roots of the quadratic equation.
From the quadratic formula, the radicand,
b2 - 4ac, will determine the Nature of the Roots.
By the nature of the roots, we mean:
whether the equation has real roots or not
if there are real roots, whether they
are different or equal
The expression b2 - 4ac is called the discriminant
of the equation ax2 + bx + c = 0 because it discriminates
among the three cases that can occur.
4.7.2

9. x2 + x - 6 = 0 The Nature of the Roots [cont’d] The value of the
discriminant is
greater than zero.
x = -3 or 2
There are two distinct
real roots.
4.7.3

10. The Nature of the Roots [cont’d]
2x2 - 3x + 8 = 0
The value of the
discriminant is
less than zero.
No real roots
4.7.5

11. The Nature of the Roots [cont’d]
We can make the following conclusions:
If b2 - 4ac > 0, then there are two different real roots.
If b2 - 4ac = 0, then there are two equal real roots.
If b2 - 4ac < 0, then there are no real roots.
Determine the nature of the roots:
x2 - 2x + 3 = 0
4x2 - 12x + 9 = 0
b2 - 4ac = (-2)2 - 4(1)(3)
= -8
b2 - 4ac = (-12)2 - 4(4)(9)
= 0
There are two equal
real roots.
The equation has no
real roots.
4.7.6

12. Determine the value of k for which the equation x2 + kx + 4 = 0 has
a) equal roots b) two distinct real roots c) no real roots
a) For equal roots, b2 - 4ac = 0.
Therefore, k2 - 4(1)(4) = 0
k = 0
k2 = 16
k = + 4
The equation has equal roots
when k = 4 and k = -4.
b) For two different real roots, b2 - 4ac > 0.
k > 0
k2 > 16
Therefore, k > 4 or k< -4. This may be written as | k | > 4.
Which numbers when squared
produce a result that is greater
than 16?
Which numbers when squared
produce a result that is less
than 16?
c) For no real roots, b2 - 4ac < 0.
k2 < 16
Therefore, -4 < k < 4. This may be written as | k | < 4.

13. Practice Time: Worksheet
Extra Homework Practice: pg. 254 #1-7