The Discriminant of a Quadratic

¨ Objective:

¤ Students will be able to (SWBAT) use the discriminant, in order to (IOT) analyze the nature of the complex solutions to quadratic equations with real coefficients.

¨ Standards:

¤ MGSE9-12.N.CN.7 – Solve Quadratics. – Solve quadratic equations with real coefficients that have complex solutions by (but not limited to) square roots, completing the square, and the quadratic formula.

The Discriminant of a Quadratic

BELL RINGER

In physics class, Tara discovers that the behavior of electrical power, !, in a particular circuit can be represented by the function "(!) = !2 + 2! + 7. If "(!) = 0, solve the equation and express your answer in simplest * + +, form.

¨ The roots of a quadratic can be found using! = #$± $&#'()*( if +, ,,

and - are all rational numbers.¨ The quantity under the square root, ,* − 4+-, truly dictates what type

of numbers the roots of a quadratic (and its x-intercepts) turn out to be.

The Discriminant of a Quadratic

The Discriminant of a Quadratic

1. For each of the following quadratics, calculate its discriminant, its roots, and state the number and nature (whether they are rational, irrational or imaginary) of the roots.(a) Case I – The Discriminant is a Perfect Square: !" + 3! − 10 = 0.) = *" − 4,- Roots: Number and Nature:

( ) ( )23 4 1 10

9 40

49

= - -

= +

=

3 49 3 72 2

5 or 2

x

x

- ± - ±= =

= -

Two unequal, rational roots.

The Discriminant of a Quadratic

1. For each of the following quadratics, calculate its discriminant, its roots, and state the number and nature (whether they are rational, irrational or imaginary) of the roots.(b) Case II – The Discriminant is Not a Perfect Square:

!" − 6! + 7 = 0.) = *" − 4,- Roots: Number and Nature:

( ) ( ) ( )26 4 1 7

36 28

8

= - -

= -

=

( )6 8 6 2 22 2

3 2

x

x

- - ± ±= =

= ±

Two unequal, irrational roots.

The Discriminant of a Quadratic

1. For each of the following quadratics, calculate its discriminant, its roots, and state the number and nature (whether they are rational, irrational or imaginary) of the roots.(c) Case III – The Discriminant is Equal to Zero: !" − 10! + 25 = 0.* = +" − 4-. Roots: Number and Nature:

( ) ( ) ( )210 4 1 25

100 100

0

= - -

= -

=

( )10 0 10 02 2

5 or 5

x

x

- - ± ±= =

=

Two equal, rational

roots or one “double” , rational root.

The Discriminant of a Quadratic

1. For each of the following quadratics, calculate its discriminant, its roots, and state the number and nature (whether they are rational, irrational or imaginary) of the roots.(d) Case IV – The Discriminant is Less than Zero: !" − 8! + 20 = 0.) = *" − 4,- Roots: Number and Nature:

( ) ( ) ( )28 4 1 20

64 80

16

= - -

= -

= -

( )8 16 8 42 2

4 2

ix

x i

- - ± - ±= =

= ±

Two unequal, imaginary roots.

The Discriminant of a Quadratic

¨ In the last lesson, we explored Case IV extensively. In the case where the discriminant is negative, the roots of the quadratic are imaginaryand it does not have x-intercepts (i.e. it does not cross the x-axis).

2. By using only the discriminant, determine the number and nature of the roots of each of the following quadratics.(a) 2"# + 7" − 4 = 0 (b) "# − 8" + 25 = 0 (c) 4"# + 4" + 1 = 0

( ) ( )2 24 7 4 2 4

49 32 81

b ac- = - -

= + =

Two unequal, rational roots.

( ) ( ) ( )22 4 8 4 1 25

64 100 36

b ac- = - -

= - = -

Two unequal, imaginary roots.

( ) ( )2 24 4 4 4 1

16 16 0

b ac- = -

= - =One double, rational root.

The Discriminant of a Quadratic

¨ 3. Consider the quadratic function whose equation ! = #$ − 4# + 4. Determine the number of x-intercepts of this quadratic from the value of its discriminant. Then, sketch its graph on the axes given. We say that this parabola is tangent to the x-axis.

( ) ( ) ( )22 4 4 4 1 4

16 16 0

b ac- = - -

= - =

One double, rational root.

The Discriminant of a Quadratic

Academic Agenda:¨ Complete the Conditional Probability Assignment.¨ Closing Question10. Consider the parabola whose equation is ! = #$ − 4 and the line whose equation is ! = 2# + ), where ) is some unknown constant. Determine the value of ) such that the line and the parabola will intersect at exactly one location. Then, sketch the system of equations on the axes below. Label their intersection point.¨ Homework: Review class notes and finish the Conditional Probability

Assignment.

The Discriminant of a Quadratic

¨ Closing Question

10. Consider the parabola whose equation is ! = #$ − 4 and the line whose equation is ! = 2# + ), where ) is some unknown constant. Determine the value of ) such that the line and the parabola will intersect at exactly one location. Then, sketch the system of equations on the axes below. Label their intersection point.

The Discriminant of a Quadratic

¨ Closing Question

( ) ( ) ( )

2

2

2

4 2

6 0 (Note in this case )

6 4 1 0

36 4 0 4 36

9

x x x b

x x b c b

b

b b

b

- = +

- - = = -

- - - =

+ = Þ = -

= -