nature of the roots and discriminant

Prepared by:Maricel T. Mas

Lipay High School

Strategic Intervention Material

in Mathematics-IX

The Nature of the Roots and The Discriminant

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.

The Standard Form of Quadratic Equation is…

ax2 + bx + c = 0

The Quadratic Formula is…

2 42

b b acxa

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.

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.

???

How to find the discriminant?

Example 1: Find the discriminant of x2 – 2x – 15 = 0

Step 2: Identify the value of a, b and c a = 1 b = -2 c = -15Step 3: Substitute these values to b2 – 4ac

Step 1: Write first the equation into standard form

Solution:D = b2 – 4ac D = (-2) 2 – 4(1)(15)D = 64

Activity No. 1.a : Set Me To Your StandardNow it’s your turn

Directions: Rewrite each quadratic equation in standard form.

1 x2 – 5x = 14

2. 2x2 + x = 5

3. x2 + 25 = 10x

4. 4x2 = 9x - 7

5. 3x2 + 2x = 5

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. x2 – 5x – 14 = 0

2. 2x2 + x = 5

3. x2 + 25 = 10x

4. 4x2 – 9x + 7 = 0

5. 3x2 + 2x - 5 = 0

a = ___ b = ___ c = ___

a = ___ b = ___ c = ___

a = ___ b = ___ c = ___

a = ___ b = ___ c = ___

a = ___ b = ___ c = ___

Activity No. 2

Directions: Using the values of a, b, and c of Activity No. 1, find the discriminant of the following using b2 – 4ac:

1. x2 – 5x – 14 = 0

2. 2x2 + x = 5

3. x2 + 25 = 10x

4. 4x2 – 9x + 7 = 0

5. 3x2 + 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

Let’s evaluate the following equations.

1. x2 – 5x – 14 = 0What 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 = 0What number is under the radical when simplified?

D= 41b2 – 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 = –31b2 – 4ac < 0, (negative)

The nature of the roots:imaginary

3.) x2 – 10x + 25 = 0

What number is under the radical when simplified?

D = 0b2 – 4ac = 0

The nature of the roots: REAL, RATIONAL, EQUAL

Determine whether the given discriminant is

a)greater than zero, perfect squareb) Greater than zero, not a perfect

squarec) Equals zerod) Less than zero

____1) 95

____2) 225

____3) -9

____4) 0

____5) 63

Activity No. 3

Activity # 4

Determine whether the given discriminant is

a) real, rational, equalb) real, rational, unequalc) real, irrational, unequald) imaginary

____1) 12

____2) 0

____3) 49

____4) -5

____1) 27

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) x2 + 14x + 49 = 0

____2) . x2 + 5x – 2 = 0

____3) 3x2 + 8x + 11 = 0

____4) x2 + 5x – 24 = 0

D=____, ____________________

D=____, ____________________D=____, ____________________

D=____, ____________________

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


3. 3x2 + 12x + 12 = 0

I. a = ___ b = ___ c = ___

II. __) 4 __) 0 __ ) -4


4. 8x2 - 9x + 11 = 0

I. a = ___ b = ___ c = ___

II. __) -172 __) -721 __ ) -271


Enrichment:

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

Answer Card

Activity No. 1.a

1. x2 – 5x – 14 =02. 2x2 + x – 5 = 03. x2 -10x + 25 = 04. 4x2 – 9x + 7 = 05. 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. 21. a. 812. c. 413. a. 04. a. -315. c. 64

Activity No. 3 1. b2. a3. d4. c5. b

Activity No. 41. c2. a3. b4. d5. b

Activity No. 51. D=0,real, rational, equal2. D= 33, real, irrational,

unequal3. D= -68, imaginary4. D= 121, real, rational,

unequal

1.) I. a=1 b= -8 c=15 II. 4 III. real, rational, unequal2.) I. a= 2 b = 4 c = 4 II. -16 III. imaginary3.) I. a= 3 b= 12 c= 12 II. 0 III. real, rational, unequal4.) I. a= 8 b= -9 c= 11 II. -271 III. imaginary

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.

nature of the roots and discriminant

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