Monday May 6th 2013

Honors Algebra

May 6th The Quadratic Formula

Why didn’t we learn this first ?

a2

Quadratic Formula

There once a negative boy who was all mixed up so he went to this radical party. Because the boy was square, he lost out on 4 awesome chicks so he cried his way home when it was all over at 2 AM.

b 2b ac4x

Quadratic Formula

a

acbbx

2

42

02 cbxax

IF

THEN

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 4

2

b b acx

a

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:

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.

WHAT THE DISCRIMINANT TELLS YOU!

Value of the Discriminant

Nature of the Solutions

Negative 2 imaginary solutions

Zero 1 Real SolutionPositive – perfect

square2 Reals- Rational

Positive – non-perfect square

2 Reals- Irrational

Understanding the discriminantDiscriminant

acb 42 # of real roots

042 acb 2 real roots

042 acb 1 real roots

042 acb No real roots

#1 Solve using the quadratic formula.0273 2 xx

a

acbbx

2

42

2 ,7 ,3 cba

)3(2

)2)(3(4)7()7( 2 x

6

24497 x

6

57 x

6

12x

6

2x

3

1 ,2x

6

257 x

#2 Solve using the quadratic formula542 2 xx

a

acbbx

2

42

5 ,4 ,2 cba

)2(2

)5)(2(4)4()4( 2 x

4

40164 x

4

564 x

4

1424 x

0542 2 xx

2

141x

9.2x 9.0x

4

1444 x

Solve using the quadratic formula3.

4.

5.

752 xx

mm 1083 2

ix2

3

2

5

3

2 ,4 m

xx 16642 8x

#3 Solve using the quadratic formula 752 xx

a

acbbx

2

42

7 ,5 ,1 cba

)1(2

)7)(1(455 2 x

2

28255 x

2

35 x

ix2

3

2

5

0752 xx

ix2

3

2

5

#4 Solve using the quadratic formula mm 1083 2

a

acbbm

2

42

8 ,10 ,3 cba

)3(2

)8)(3(4)10()10( 2 m

6

9610010 m

6

1410 m

6

24m

6

4m

3

2 ,4 m

08103 2 mm

6

19610 m

#5 Solve using the quadratic formula xx 16642

a

acbbx

2

42

64 ,16 ,1 cba

)1(2

)64)(1(41616 2 x

2

25625616 x

2

016 x

8x

064162 xx

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

1. 9,7

2.(6, 14)

3. 3,8

7 34.

2

5 475.

6

i

i

The Discriminant

Discriminant In the quadratic formula, the expression underneath the radical that describes the nature of the roots.

Understanding the discriminantDiscriminant

acb 42 # of real roots

042 acb 2 real roots

042 acb 1 real roots

042 acb No real roots

#6 Using the discriminant0134 2 yy

acb 4 nt discrimina 2

1 ,3 ,4 cba

)1)(4(4)3( nt discrimina 2

169 nt discrimina

52 nt discrimina

052

roots real 2

0134 2 yy

542 2 xx

484 2 xx

1)2)3)

#7 Using the discriminant54 2 xx

acb 4 nt discrimina 2

5 ,1 ,4 cba

)5)(4(4)1( nt discrimina 2

801 nt discrimina

79 nt discrimina

079

roots real 0

Using the discriminant

8.

9.

542 2 xx

484 2 xx

56nt discrimina

roots real 2

0 nt discrimina

root real 1

#8 Using the discriminant

542 2 xx

acb 4 nt discrimina 2

0542 2 xx

)5)(2(4)4( nt discrimina 2

4061 nt discrimina

56nt discrimina

056

roots real 2

#9 Using the discriminant484 2 xx

acb 4 nt discrimina 2

0484 2 xx

)4)(4(4)8( nt discrimina 2

6446 nt discrimina

0 nt discrimina

00

root real 1