6: Discriminant6: Discriminant

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

Discriminant

Module C1

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Discriminant

Because we square root the discriminant, we get different types of roots depending on its sign.

The Discriminant of a Quadratic FunctionThe formula for solving a quadratic equation is

a

acbbx

2

42

The part is called the discriminant

acb 42

Discriminant

452 xxy

we consider the graph of the function 452 xx

To investigate the roots of the equation

452 xx 0

yy

0

The roots of the equation are at the points where y = 0

The roots are real and distinct.( different )

1625 The discriminantacb 42

9 0

The Discriminant of a Quadratic Function

( x = 1 and x = 4)

Discriminant

For the equation

0442 xxthe discriminant

acb 42 1616

442 xxy

The roots are real and equal

0

The Discriminant of a Quadratic Function

( x = 2)

Discriminant

For the equation . . . 0742 xx

. . . the discriminant

acb 42 12

There are no real roots as the function is never equal to zero

2816

The Discriminant of a Quadratic Function

If we try to solve , we get0742 xx

2

124 x

The square of any real number is positive so there are no real solutions to 12

742 xxy0

Discriminant

042 acb

042 acb

The part is called the discriminant

acb 42

042 acb The roots are real and equalThe roots are not real

The roots are real and distinct( different )

SUMMARY

The formula for solving the quadratic equation

a

acbbx

2

42 02 cbxax is

If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

Discriminant

1 (a) Use the discriminant to determine the nature of the roots of the following quadratic equations:

0114442 ))((acb

0122 xx(i)

(ii)

0122 xx

(b) Check your answers by completing the square to find the vertex of the function and sketching.

Solution: (a) (i) The roots are real and

equal.(ii) 844114442 ))((acb

The roots are real and distinct.

Exercise

Discriminant

122 xxy

122 xxy

(b) Check your answers by completing the square to find the vertex of the function and sketching.(b)

(i)

122 xx21)( x

Vertex is ( -1,0 )

(ii)

122 xx

21 2 )(x

Roots of equation(real and equal)

Roots of equation(real and distinct)

Vertex is ( 1,-2 )

Discriminant

2. Determine the nature of the roots of the following quadratic equations ( real and distinct or real and equal or not real ) by using the discriminant. DON’T solve the equations.(a) 0962 xx

(c) 0295 2 xx

(b) 0952 2 xx

363691464 22 ))(()(acbRoots are real and equal

121408125494 22 ))(()(acb

47722592454 22 ))(()(acbThere are no real roots

Roots are real and distinct

0

0

0

Discriminant

Discriminant

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Roots, Surds and Discriminant

Roots is just another word for solutions !e.g. Find the roots of the equation

0122 xx

Completing the square:

2

11422 2 ))(( x

a

acbbx

2

42

Solution: There are no factors, so we can either complete the square or use the quadratic formula. 0111 2 )(x

21 )(x 21 x

Using the formula:

2

82 x

(1)

(2)

Roots, Surds and Discriminant

2

212 )( x

The answers from the quadratic formula can be simplified:

2

82 xWe

have

248 However,

2224

4 is a perfect square so can be square-rooted, so

2

82 xSo,

2

222 x

2 is a common factor of the numerator, so

21 x

Numbers such as are called surds

8

We have simplified the surd

1

1

Roots, Surds and Discriminant

042 acb

042 acb

042 acb

The part is called the discriminantacb 42

The roots are real and equal.

The roots are not real.

The roots are real and distinct.( different )

The Discriminant

The formula for solving the quadratic equation

a

acbbx

2

42 02 cbxax is

If we try to solve an equation with no real roots, we will be faced with the square root of a negative number!

Roots, Surds and Discriminant

452 xxy

The roots are real and distinct.( different )

1625 The discriminant

acb 42 90

e.g. For

452 xxy

Roots, Surds and Discriminant

acb 42 1616 0

The discriminant

e.g. For 442 xxy

442 xxy

The roots are real and equal.

Roots, Surds and Discriminant

742 xxy

acb 42 12

There are no real roots as the function is never equal to zero.

2816

0

The discriminant

e.g. For 742 xxy