nature of the roots of quadratic equation and the discriminant

1. Prepared by:
Maricel T. Mas
Lipay High School
Strategic Intervention Material
in Mathematics-IX
The Nature of the Roots
and The Discriminant

2. Guide Card
Least Mastered Skill:
• Identify the Nature of the Roots
Sub tasks:
 Identify values of a, b and c of a quadratic
equation,
 Find the discriminant; and
 Describe the nature of roots of quadratic
equation.

3. The Standard Form of
Quadratic Equation is…
ax2 + bx + c = 0
The Quadratic Formula is…
2
4
2
b b ac
x
a
  


4. WHY USE THE
QUADRATIC FORMULA?
 The quadratic formula allows you to solve ANY quadratic
equation, even if you cannot factor it.
 An important piece of the quadratic formula is what’s
under the radical:
b2 – 4ac
 This piece is called the discriminant.

5. WHY IS THE DISCRIMINANT
IMPORTANT?
The discriminant tells you the number and types of answers
(roots) you will get. The discriminant can be +, –, or 0
which actually tells you a lot! Since the discriminant is
under a radical, think about what it means if you have
a positive or negative number or 0 under the radical.
???

6. How to find the discriminant?
Example 1: Find the discriminant of
x
2
– 2x – 15 = 0
Step 2: Identify the value of a, b and c
a = 1 b = -2 c = -15
Step 3: Substitute these values to b
2
– 4ac
Step 1: Write first the
equation into
standard form
Solution:
D = b
2
– 4ac
D = (-2)
2
– 4(1)(15)
D = 64

7. Activity No. 1.a : Set Me To Your Standard
Now it’s your turn
Directions: Rewrite each quadratic equation in standard form.
1 x
2
– 5x = 14
2. 2x
2
+ x = 5
3. x
2
+ 25 = 10x
4. 4x
2
= 9x - 7
5. 3x
2
+ 2x = 5

8. Activity No. 1.b
Now it’s your turn
Directions: Using the given quadratic equations on activity no
1.b, identify the values of a, b, and c.
1. x
2
– 5x – 14 = 0
2. 2x
2
+ x = 5
3. x
2
+ 25 = 10x
4. 4x
2
– 9x + 7 = 0
5. 3x
2
+ 2x - 5 = 0
a = ___ b = ___ c = ___
a = ___ b = ___ c = ___
a = ___ b = ___ c = ___
a = ___ b = ___ c = ___
a = ___ b = ___ c = ___

9. Activity No. 2
Directions: Using the values of a, b, and c of Activity No. 1, find the discriminant
of the following using b
2
– 4ac:
1. x
2
– 5x – 14 = 0
2. 2x
2
+ x = 5
3. x
2
+ 25 = 10x
4. 4x
2
– 9x + 7 = 0
5. 3x
2
+ 2x - 5 = 0
a. 81 b. 11 c. -31
a. 39 b. - 39 c. 41
a. 0 b. 1 c. 100
a. - 31 b. 31 c. 81
a. -56 b. -64 c. 64

10. Let’s evaluate the
following equations.
1. x2
– 5x – 14 = 0
What number is under the radical
when simplified?
D=81
b2
– 4ac > 0, perfect square
 The nature of the roots :
REAL, RATIONAL, UNEQUAL
2. ) 2x2
+ x – 5 = 0
What number is under the
radical when simplified?
D= 41
b2
– 4ac > 0, not a perfect
square
The nature of the roots:
REAL, IRRATIONAL, UNEQUAL
4.) 4x2
– 9x + 7 = 0
What number is under the
radical when simplified?
D = –31
b2
– 4ac < 0, (negative)
The nature of the roots:
imaginary
3.) x2
– 10x + 25 = 0
What number is under the
radical when simplified?
D = 0
b2
– 4ac = 0
The nature of the roots:
REAL, RATIONAL, EQUAL

11. Determine whether the given discriminant is
a)greater than zero, perfect square
b) Greater than zero, not a perfect
square
c) Equals zero
d) Less than zero
____1) 95
____2) 225
____3) -9
____4) 0
____5) 63
Activity No. 3

12. Activity # 4
Determine whether the given discriminant is
a) real, rational, equal
b) real, rational, unequal
c) real, irrational, unequal
d) imaginary
____1) 12
____2) 0
____3) 49
____4) -5
____1) 27

13. Activity No. 5: Try These.
For each of the following quadratic equations,
a) Find the value of the discriminant, and
b) Describe the number and type of roots.
____1) x
2
+ 14x + 49 = 0
____2) . x
2
+ 5x – 2 = 0
____3) 3x
2
+ 8x + 11 = 0
____4) x
2
+ 5x – 24 = 0
D=____, ____________________
D=____, ____________________D=____, ____________________
D=____, ____________________

14. Assessment Card No. 1:
Write the values of a, b & c in the quadratic equation, then check the
discriminant and nature of roots of quadratic equation .
1. x2 – 8x + 15 = 0
I. a = ___ b = ___ c = ___
II. __ 4 __) 0 __ ) -4
__real, rational, equal
__real, rational, unequal
__real, irrational, unequal
__imaginary
2. 2x2 + 4x + 4 = 0
I. a = ___ b = ___ c = ___
II. __) 16 __) 0 __ ) -16
__real, rational, equal
__real, rational, unequal
__real, irrational, unequal
__imaginary

15. 3. 3x2 + 12x + 12 = 0
I. a = ___ b = ___ c = ___
II. __) 4 __) 0 __ ) -4
__real, rational, equal
__real, rational, unequal
__real, irrational, unequal
__imaginary
4. 8x2 - 9x + 11 = 0
I. a = ___ b = ___ c = ___
II. __) -172 __) -721 __ ) -271
__real, rational, equal
__real, rational, unequal
__real, irrational, unequal
__imaginary

16. Enrichment:
Directions: Determine the nature of the roots of the following
quadratic equations.

17. Answer Card
Activity No. 1.a
1. x2 – 5x – 14 =0
2. 2x2 + x – 5 = 0
3. x2 -10x + 25 = 0
4. 4x2 – 9x + 7 = 0
5. 3x2 + 2x – 5 = 0
Activity No. 1.b.
1. a = 1 b = -5 c=-14
2. a = 2 b = 1 c = -5
3. a = 1 b = -10 c = 25
4. a = 4 b = -9 c = 7
5. a = 3 b = 2 c = -5
Activity No. 2
1. a. 81
2. c. 41
3. a. 0
4. a. -31
5. c. 64
Activity No. 3
1. b
2. a
3. d
4. c
5. b
Activity No. 4
1. c
2. a
3. b
4. d
5. b
Activity No. 5
1. D=0,real, rational, equal
2. D= 33, real, irrational,
unequal
3. D= -68, imaginary
4. D= 121, real, rational,
unequal
1.) I. a=1 b= -8 c=15
II. 4
III. real, rational, unequal
2.) I. a= 2 b = 4 c = 4
II. -16
III. imaginary
3.) I. a= 3 b= 12 c= 12
II. 0
III. real, rational, unequal
4.) I. a= 8 b= -9 c= 11
II. -271
III. imaginary

18. References
Jose-Dilao, Soledad, Orines, and Bernabe,
Julieta G. Advanced Algebra, Trigonometry
and Statistics IV, SD Publications, Inc, 2009, p.
73
Learner’s Material Mathematics – Grade 9
First Edition, 2014 pp. 65-70.