This powerpoint presentation is all about discriminants. This will surely help Math students for their future lessons.

1. Discriminan
t of a
Quadratic
Equation

2. OBJECTIVES
• characterizes the roots of a quadratic
equation using the discriminant.
• describes the relationship between
the coefficients and the roots of a
quadratic equation.

3. 𝒙 =
−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄
𝟐𝒂
Recall that the roots of the
quadratic equation 𝒂𝒙𝟐
+ 𝒃𝒙 +
𝒄 = 𝟎 are given by the quadratic
formula:

4. We can see that the radicand
𝒃𝟐 − 𝟒𝒂𝒄 determines the nature
of these roots. This radicand is
called the discriminant of the
quadratic equation.

5. If: Then the roots are:
𝐷 = 0 Real, rational, and
equal
𝐷 > 0
𝑎𝑛𝑑 𝑖𝑠 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒
Real, rational, and
unequal
𝐷 > 0
𝑎𝑛𝑑 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒
Real, irrational, and
unequal
𝐷 < 0 Not real

6. A. 𝒙𝟐
− 𝟖𝒙 + 𝟏𝟔 = 𝟎
B. 𝟐𝒙𝟐
− 𝟓𝒙 − 𝟑 = 𝟎
C. 𝒙𝟐
+ 𝟓𝒙 + 𝟑 = 𝟎
D. 𝒙𝟐
− 𝒙 + 𝟐 = 𝟎
EXAMPLE

7. 𝒙𝟐
+ 𝟑𝒙 + 𝟏 = 𝟎
Directions: Determine the nature of
the roots of the following quadratic
equations using the discriminant.
𝒙𝟐
+ 𝟕𝒙 + 𝟔 = 𝟎
𝒙𝟐
− 𝟏𝟎𝒙 + 𝟏𝟐 = 𝟎
𝒙𝟐
+ 𝟒𝒙 − 𝟏𝟐 = 𝟎
𝟐𝒙𝟐
− 𝟕𝒙 + 𝟔 = 𝟎

8. Seatwork

9. If 𝒙𝟏 and 𝒙𝟐 are the roots of the
quadratic equation 𝒂𝒙𝟐
+ 𝒃𝒙 + 𝒄 = 𝟎,
then
Sum of the Roots: 𝒙𝟏 + 𝒙𝟐 =
−𝒃
𝒂
Product of the Roots:𝒙𝟏𝒙𝟐 =
𝒄
𝒂
Relationship between the
coefficients and Roots of a
Quadratic Equation

10. Take note that if the general quadratic
equation is written in the form 𝒙𝟐
+
𝒃
𝒂
𝒙 +
𝒄
𝒂
= 𝟎 , then 𝒙𝟐 −
𝒔𝒖𝒎 𝒐𝒇 𝒓𝒐𝒐𝒕𝒔 𝒙 +
𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒓𝒐𝒐𝒕𝒔 = 𝟎.
This relationship is useful in writing
quadratic equations whose roots are
given.
Relationship between the
coefficients and Roots of a
Quadratic Equation

11. EXAMPLE
A. Find the sum and product
of the roots of 𝟐𝒙𝟐
+ 𝟖𝒙 − 𝟏𝟎
= 𝟎.
B. Find the sum and product
of the roots of 𝒙𝟐
+ 𝟕𝒙 − 𝟏𝟖
= 𝟎.

12. EXAMPLE
A. Write a quadratic equation
whose roots are 𝟐 and −𝟓.
B. Write a quadratic equation
whose roots are 𝟓 + 𝟑 and
𝟓 − 𝟑 .

13. 5 𝑎𝑛𝑑 2
Directions: Write a quadratic
equation given the following roots.
1
4
2
5
3
−8 𝑎𝑛𝑑 − 10
4 𝑎𝑛𝑑 7
−3 𝑎𝑛𝑑 15
1 𝑎𝑛𝑑 − 6