The Quadratic Formula and the Discriminant

2 cxay x b

The Discriminant

2 4

2

b b acx

a

The DiscriminantThe Discriminant is:

Way to solve: Number and kind of solutions

Positive and Square

Factor 2 Rational

Zero Factor 1 Rational

Positive not square

Quadratic formula or Complete the square

2 Irrational

Negative Don’t solve Imaginary Solutions

2 4 0b ac

2 Real Solutions

Picture: Positive Discriminant

2 4 0b ac

1 Real Solution

Picture: Zero Discriminant

2 4 0b ac 0 Real Solution. 2 Complex Solutions.

Picture: Negative Discriminant

Using the discriminant

0134 2 xx

acb 4 nt discrimina 2

1 ,3 ,4 cba

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

169 nt discrimina

52 nt discrimina roots real 2

Factor

Using the discriminant

14

1

01014

0)1)(14(

0134 2

xorx

xorx

xx

xx

Using the discriminant

54 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 solvenot do

Using the discriminant

484 2 xx

acb 4 nt discrimina 2

0484 2 xx

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

6446 nt discrimina

0 nt discrimina

root real 1 factor

Using the discriminant

1

01

0)1)(1(4

0)12(4

0484

484

2

2

2

x

x

xx

xx

xx

xx

Using the discriminant

9183 2 xx

acb 4 nt discrimina 2

09183 2 xx

)9)(3(4)18( nt discrimina 2

108324 nt discrimina

216 nt discrimina

roots irrational 2

formula quadraticor square thecomplete

Using the discriminant

63

63

6)3(

9396

36

9183

2

2

2

2

x

x

x

xx

xx

xx

Worksheet

If the discriminant is positive and not square alternate complete the square and

quadratic formula.