Roots of Quadratic Equation

2. ROOTS OF
QUADRATIC
EQUATIONS
MARK JOVEN A. ALAM-ALAM, LPT

3. QUADRATIC FORMULA
X =
−𝑏 ± 𝑏2−4𝑎𝑐
2𝑎
discriminant of the quadratic
equation
 𝑏2
− 4𝑎𝑐
determines the number and type of

4. NATURE OF ROOTS OF QUADRATIC
EQUATIONS
In the quadratic equation 𝒂𝒙𝟐
+ bx + c = 0, where a≠
0:
a. If 𝒃𝟐
− 𝟒𝒂𝒄 = 0
Roots are two equal rational numbers
b. If 𝒃𝟐
− 𝟒𝒂𝒄 > 0, and perfect square
Roots are two unequal rational numbers
c. If 𝒃𝟐
− 𝟒𝒂𝒄 > 0, and not a perfect square
Roots are two unequal irrational numbers
d. If 𝒃𝟐
− 𝟒𝒂𝒄 < 0
Roots are two imaginary numbers

5. Example 1 : Without solving for the roots, tell
the nature of the roots of the given quadratic
equations.
a. 2𝑥2 + 5x – 7 = 0
b. 7𝑥2
= 8x – 10
c. 3𝑥2 – 1= 12x

6. SOLUTION:
a. 2𝑥2
+ 5x – 7 = 0
a = 2 b= 5 c=-
7
𝑏2
− 4𝑎𝑐 = 52
- 4(2)(-7)
= 25 + 56
= 81
 Roots are two
unequal rational
numbers.

7. SOLUTION:
b. 7𝑥2
= 8x – 10
7𝑥2
- 8x + 10 = 0
a = 7 b= -8 c=10
𝑏2
− 4𝑎𝑐 = (−8)2
-
4(7)(10)
= 64 - 280
 Roots are two
imaginary
numbers.

8. SOLUTION:
c. 3𝑥2
– 1= 12x
3𝑥2
– 12x - 1= 0
a = 3 b= -12 c=-1
𝑏2
− 4𝑎𝑐 = (−12)2
-
4(3)(-1)
= 144 + 12
 Roots are two
unequal irrational
numbers.

9. SUM AND PRODUCT OF THE ROOTS OF
QUADRATIC EQUATIONS
SUM OF ROOTS
𝑟1 + 𝑟2 = -
𝑏
𝑎
PRODUCT OF
ROOTS
𝑟1 𝑟2 =
𝑐
𝑎

10. Example 2 : Without solving for the roots,
determine the sum and product of the roots
of the following equations.
a. 𝑥2
+ 7x + 12 = 0
b. 2𝑥2 = 18
c. 4𝑥2 – 2x +
1
4
= 0

11. SOLUTION:
a. 𝑥2
+ 7x + 12 = 0
a = 1 b= 7
c=12
𝑟1 + 𝑟2 = -
𝑏
𝑎
= -
7
1
= - 7
𝑟1 𝑟2 =
𝑐
𝑎
=
12
1
= 12

12. SOLUTION:b. 2𝑥2
= 18
2𝑥2
- 18 = 0
a = 2 b= 0 c=-18
𝑟1 + 𝑟2 = -
𝑏
𝑎
= -
0
2
= 0
𝑟1 𝑟2 =
𝑐
𝑎
=
−18
2
= -9

13. SOLUTION:
c. 4𝑥2
– 2x +
1
4
= 0
a = 4 b= -2 c=
1
4
𝑟1 + 𝑟2 = -
𝑏
𝑎
= -
−2
4
=
1
2
𝑟1 𝑟2 =
𝑐
𝑎
=
1
4
4
=
1
4
●
1
4
=
1
16

14. DERIVING A QUADRATIC EQUATION GIVEN THE
SUM AND PRODUCT OF ITS ROOTS
𝒂𝒙𝟐
+ bx + c = 0
𝒙𝟐
+
𝑏
𝑎
x +
𝑐
𝑎
= 0
𝒙𝟐
- (-
𝑏
𝑎
)x +
𝑐
𝑎
= 0
𝒙𝟐
- (𝑟1 + 𝑟2)x + (𝑟1 𝑟2) =
0

15. Example 3 : Find a quadratic equation in
standard form whose roots have the given sum
and product, respectively.
a. 4 and -12
b. -
1
6
and -
1
6

16. SOLUTION:
a. 4 and -12
𝑟1 + 𝑟2 = 4 𝑟1 𝑟2= -12
𝒙𝟐 - (𝑟1 + 𝑟2)x + (𝑟1 𝑟2) = 0
𝒙𝟐 - (4)x + (-12) = 0
𝒙𝟐 - 4x -12 = 0

17. SOLUTION: b. -
1
6
and -
1
6
𝑟1 + 𝑟2 = -
1
6
𝑟1 𝑟2= -
1
6
𝑥2
- (𝑟1 + 𝑟2)x + (𝑟1 𝑟2) = 0
𝑥2
- (−
1
6
)x + (-
1
6
) = 0
𝑥2
+
1
6
x -
1
6
= 0
6𝑥2
+ x – 1 = 0