The Discriminant

Check for Understanding – Given a quadratic equation use the discriminant to determine the nature of the roots.

Terms we need to know:

1) Quadratic Formula2) Real number system3) Rational numbers4) Irrational numbers5) Perfect squares6) Complex numbers7) Imaginary numbers

Examples of Rational and Irrational numbersRational

a)2Irrational

Most common irrational numbers involve the √ symbol.Many square roots, cube roots, etc. are irrational.

Let’s Practice

• Record your answer for each

Complex Number SystemReals

Rationals(fractions, decimals)

Integers(…, -1, -2, 0, 1, 2, …)

Whole(0, 1, 2, …)

Natural(1, 2, …)

Irrationals(no fractions)

pi, e

Imaginary

i, 2i, -3-7i, etc.

THE QUADRATIC FORMULA

1. When you solve using completing the square on the general formula you get:

2. This is the quadratic formula!3. Just identify a, b, and c then

substitute into the formula.

2 42

b b acxa

2 0ax bx c

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

What is the discriminant?

The discriminant is the expression b2 – 4ac.

We represent the discriminant with D

The value of the discriminant can be usedto determine the number and type of rootsof a quadratic equation.

How have we previously used the discriminant?

We used the discriminant to determine whether a quadratic polynomial couldbe factored.

If the value of the discriminant for a quadratic polynomial is a perfect square, the polynomial can be factored.

Let’s put all of that information in a chart.

Value of Discriminant

Type andNumber of Roots

Sample Graphof Related Function

D > 0,D is a perfect square

D > 0,D NOT a perfect

square

D = 0

D < 0

During this presentation, we will complete a chart that shows how the value of the discriminantrelates to the number and type of roots of aquadratic equation.

Rather than simply memorizing the chart, thinkAbout the value of b2 – 4ac under a square rootand what that means in relation to the roots ofthe equation.

Solve These…

Use the quadratic formula to solve eachof the following equations?

1. x2 – 5x – 14 = 0

2. 2x2 + x – 5 = 0

3. x2 – 10x + 25 = 0

4. 4x2 – 9x + 7 = 0

Let’s evaluate the first equation.

x2 – 5x – 14 = 0

What number is under the radical when simplified?

81 The discriminant is 81; which is a perfect square

What are the solutions of the equation?

–2 and 7 which are both rational numbers.

If the value of the discriminant is positive,the equation will have 2 real roots.

If the value of the discriminant is a perfect square, the roots will be rational.

Let’s look at the second equation.

2x2 + x – 5 = 0

What number is under the radical when simplified?

41 (which is not a perfect square.)

What are the solutions of the equation?Both solutions are irrational 1 41

4

If the value of the discriminant is positive,the equation will have 2 real roots.

If the value of the discriminant is a NOTperfect square, the roots will be irrational.

Now for the third equation.

x2 – 10x + 25 = 0

What number is under the radical when simplified?

0

What are the solutions of the equation?

5 (double root)

If the value of the discriminant is zero,the equation will have 1 real, root; it willbe a double root.

If the value of the discriminant is 0, theroots will be rational.

Last but not least, the fourth equation.

4x2 – 9x + 7 = 0

What number is under the radical when simplified?

–31

What are the solutions of the equation?There are no real solutions. The solution is

imaginary. 9 318i

If the value of the discriminant is negative,the equation will have 2 complex roots;they will be complex conjugates.Not to panic if you don’t recognize these

above terms… we will cover them in the next lesson.

Let’s put all of that information in a chart.

Value of Discriminant

Type andNumber of Roots

Sample Graphof Related Function

D > 0,D is a perfect square

2 real, rational roots

D > 0,D NOT a perfect

square2 real,

Irrational roots

D = 0 1 real, rational root(double root)

D < 02 complex roots

(complex conjugates)

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 3. 3x2 + 8x + 11 = 0

2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0

The Answers

1. x2 + 14x + 49 = 0

D = 0

1 real, rational root (double root)

2. x2 + 5x – 2 = 0

D = 33 2 real, irrational roots

3. 3x2 + 8x + 11 = 0

D = –68

2 complex roots (complex conjugates)

4. x2 + 5x – 24 = 0

D = 121

2 real, rational roots

WHY IS THE DISCRIMINANT IMPORTANT?

The discriminant tells you the number and types of answers

(roots) you will get. The discriminant can be +, –, or 0which 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.

WHAT THE DISCRIMINANT TELLS YOU!

Value of the Discriminant

Nature of the Solutions

Negative 2 imaginary solutionsZero 1 Real Solution

Positive – perfect square

2 Reals- Rational

Positive – non-perfect square

2 Reals- Irrational

Example #1

22 7 11 0x x

Find the value of the discriminant and describe the nature of the roots (real,imaginary, rational, irrational) of each quadratic equation. Then solve the equation using the quadratic formula)

1. a=2, b=7, c=-11

Discriminant = 2

2

4

(7) 4(2)( 11)49137

88

b ac

Discriminant =

Value of discriminant=137Positive-NON perfect squareNature of the Roots – 2 Reals - Irrational

Example #1- continued22 7 11 0x x

2

2

42

7 7 4(2)( 11)2(

2, 7, 11

7 137 2 Reals - Irrational4

2)

a b

b aca

c

b

Solve using the Quadratic Formula

Solving Quadratic Equations by the Quadratic Formula

2

2

2

2

2

1. 2 63 0

2. 8 84 0

3. 5 24 0

4. 7 13 0

5. 3 5 6 0

x x

x x

x x

x x

x x

Try the following examples. Do your work on your paper and then check your answers.

1. 9,7

2.(6, 14)3. 3,8

7 34.2

5 475.6

i

i