Quadratic Functions (4)

•What is the discriminantWhat is the discriminant•Using the discriminantUsing the discriminant

25 =+5 or -5

1 =+1 or -1

(92) =+9 or -9

(-4) =can’t do

acb 42

In

What can we say about ...

To get a solution for

x ?

acb 42

InWhat can we say about ...

If it’s negative then it has no solutions---> cannot square root a negative number

If it’s zero then it only has only solution

The discriminant

acb 42 This is the discriminant of the equation ax2+bx+c=0

Using the discriminantThe discriminant can be used to give us important information about the roots of our quadratic.

The “roots” are basically our solutions when ax2+bx+c=0

Roots

Which is which?b2-4ac > 0b2-4ac < 0b2-4ac = 0

b2-4ac > 0

b2-4ac < 0

b2-4ac = 0

Using the discriminant

If b2-4ac > 0 Equation has two distinct roots.

If b2-4ac < 0 Equation has no real roots.

If b2-4ac = 0 Equation has repeated

roots.

How it is used - exampleCalculate the discriminant of 2x2+7x+7=0

and hence prove 2x2+7x+7 is always > 0a = [coefficient of

x2]b = [coefficient of x]c= [constant]

= 2= 7= 7

b2 - 4ac = 72 – (4 x 2 x 7) =49 - 56 = -7

If b2-4ac < 0 Equation has no real roots.

Therefore, doesn’t cross the x-axis and is always positive

How it is used - example

If b2-4ac > 0 Equation has two distinct roots.

For what values of ‘k’ does the equation 2x2 -3x+k=0 have real roots

a = [coefficient of x2]b = [coefficient of x]c= [constant]

= 2= -3= k

b2 - 4ac > 0(-3)2 – (4 x 2 x k) > 09 – 8k >09 > 8k

9/8 > kk < 9/8

Have a go

If b2-4ac < 0 Equation has no distinct roots.

For what values of ‘k’ does the equation 3x2 + 5x+k=0 have no real roots

a = [coefficient of x2]b = [coefficient of x]c= [constant]

= 3= 5= k

b2 - 4ac < 052 – (4 x 3 x k) < 025 – 12k < 025 < 12k

25/12 < kk > 25/12