Intro: Intro: We already know the standard form of a quadratic equation is: y = ax2 ax2 ax2 ax2 + bx bx + c The The constants constants are: a, b, c The.

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

y = y = aaxx22 + + bbx + x + cc•The The constantsconstants are: are: a , b, ca , b, c•The The variablesvariables are: are: y, xy, x

•The The ROOTSROOTS (or (or solutionssolutions) of a ) of a polynomial are polynomial are its its x-interceptsx-intercepts

•Recall: The Recall: The x-x-interceptsintercepts occur where occur where y y = 0= 0..

Roots

•Example:Example: Find Find the roots: the roots: y = xy = x22 + x - 6+ x - 6

•Solution:Solution: Factoring: Factoring: y = (x y = (x + 3)(x - 2)+ 3)(x - 2)

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

Roots

•But what about But what about NASTYNASTY trinomials trinomials that don’t that don’t factor?factor?

•Abu Ja'far Abu Ja'far Muhammad ibn Muhammad ibn Musa Al-KhwarizmiMusa Al-Khwarizmi

•BornBorn: about 780 in : about 780 in Baghdad (Iraq)Baghdad (Iraq)

•DiedDied: about 850: about 850

•After centuries of After centuries of work, work, mathematicians mathematicians realized that as realized that as long as you know long as you know the coefficients, the coefficients, you can find the you can find the roots of the roots of the quadratic. Even if quadratic. Even if it doesn’t factor!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±210

x=8±210

x=8+210

=1010

=1

x=8−210

=6

10=

35 Roots

y=5(1)2 −8(1)+3y=5−8+3y=0

y=5 35( )

2−8 3

5( )+3

y=5 925( )− 24

5( )+3

y= 4525( )− 24

5( )+3

y= 95( )− 24

5( )+ 155( )

y=0

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=

24

=12

x=−164

=−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:Solve: 3x 3x22 = 7 - 2x = 7 - 2x•Solution:Solution: 3x 3x22 + 2x - 7 = 0 + 2x - 7 = 0•a = 3, b = 2, c = -7a = 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± 223

We use the quadratic We use the quadratic formula to solve second formula to solve second degree equations. degree equations. Mathematicians tried Mathematicians tried for 300 years to solve for 300 years to solve higher-degree higher-degree equations until Niels equations until Niels Abel (top picture) Abel (top picture) proved that no formula proved that no formula can be used to solve all can be used to solve all fifth-degree equations. fifth-degree equations. He was 22!He was 22!

• Evariste Galois (bottom Evariste Galois (bottom picture) showed that picture) showed that there is no universal there is no universal formula for any equations formula for any equations higher than the fourth higher than the fourth degree. When Galois was degree. When Galois was 20, he wrote in ONE 20, he wrote in ONE NIGHT much of the basis NIGHT much of the basis for a new theory of for a new theory of solving equations. Sadly, solving equations. Sadly, he was killed in a duel he was killed in a duel the next day.the next day.

• MORAL: Don’t do your MORAL: Don’t do your homework late at night. homework late at night.