BY: NASIRA ATEEQFROM: D A PUBLIC SCHOOL

(O & A LEVELS)

Intro: We already know the standard form of a quadratic equation is:

y = ax2 + bx + c The constants are: a , b, c The variables are: y, x

The ROOTS (or solutions) of a polynomial are its x-intercepts

Recall: The x-intercepts occur where y = 0.

Roots

Example: Find the roots:

y = x2 + x - 6Solution: Factoring:

y = (x + 3)(x - 2) 0 = (x + 3)(x - 2)The roots are:

x = -3; x = 2

Roots

But what about NASTY trinomials that don’t factor?

Abu Ja'far Muhammad ibn Musa Al- Khwarizmi

Born: about 780 in Baghdad (Iraq)

Died: about 850

After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor!

y ax2 bx c, a 0

x b b2 4ac

2a

Solve: y = 5x2 8x 3

x b b2 4ac

2a

a 5, b 8, c 3

x ( 8) ( 8)2 4(5)(3)

2(5)

x 8 64 60

10

x 8 4

10

x 8 2

10

x 8 2

10

x 8 2

10

10

101

x 8 2

10

6

10

3

5Roots

y 5(1)2 8(1) 3y 5 8 3y 0

y 5 35 2

8 35 3

y 5 925 24

5 3

y 4525 24

5 3

y 95 24

5 155

0y

Plug in your Plug in your answers for answers for xx..If you’re right, If you’re right, you’ll get you’ll get y = y = 00..

Solve : y 2x2 7x 4

a 2, b 7, c 4

x b b2 4ac

2a

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

2(2)

x 7 49 32

4

x 7 81

4

x 7 9

4x

2

4

1

2

x 16

4 4

RememberRemember: All the terms must : All the terms must be on be on one sideone side BEFORE you use BEFORE you use the quadratic formula.the quadratic formula.

•Example:Example: Solve Solve 3m3m22 - 8 = 10m - 8 = 10m

•Solution:Solution: 3m3m2 2 - 10m - 8 = 0- 10m - 8 = 0

•a = 3, b = -10, c = -8a = 3, b = -10, c = -8

Solve: 3x2 = 7 - 2xSolution: 3x2 + 2x - 7 = 0a = 3, b = 2, c = -7

x b b2 4ac

2a

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

2(3)

x 2 4 84

6

x 2 88

6

x 2 4 • 22

6

x 2 2 22

6x

1 22

3