Cambridge Additional Mathematics

Quadratics (Chapter 3) 73

In the quadratic formula, the quantity b^2 ¡ 4 ac under the square root sign is called thediscriminant.

The symboldelta¢is used to represent the discriminant, so ¢=b^2 ¡ 4 ac.

The quadratic formula becomes x=

¡b§

p

¢

2 a

where¢replaces b^2 ¡ 4 ac.

² If ¢=0, x=

¡b

2 a

is theonly solution(arepeatedordouble root)

² If ¢> 0 ,

p

¢is a positive real number, so there aretwo distinct real roots

x=

¡b+

p

¢

2 a

and x=

¡b¡

p

¢

2 a

² If ¢< 0 ,

p

¢is not a real number and so there areno real roots.

² Ifa,b, andcare rational and¢is asquarethen the equation has two rational roots which can be

found by factorisation.

Example 9 Self Tutor

Use the discriminant to determine the nature of the roots of:

a 2 x^2 ¡ 2 x+3=0 b 3 x^2 ¡ 4 x¡2=0

a ¢=b^2 ¡ 4 ac

=(¡2)^2 ¡4(2)(3)

=¡ 20

Since ¢< 0 , there are no real roots.

b ¢=b^2 ¡ 4 ac

=(¡4)^2 ¡4(3)(¡2)

=40

Since ¢> 0 , but 40 is not a square,

there are 2 distinct irrational roots.

Example 10 Self Tutor

Consider x^2 ¡ 2 x+m=0. Find the discriminant¢, and hence find the values ofmfor which

the equation has:

a a repeated root b 2 distinct real roots c no real roots.

x^2 ¡ 2 x+m=0 has a=1, b=¡ 2 , and c=m

) ¢=b^2 ¡ 4 ac

=(¡2)^2 ¡4(1)(m)

=4¡ 4 m

a For a repeated root

¢=0

) 4 ¡ 4 m=0

) 4=4m

) m=1

b For 2 distinct real roots

¢> 0

) 4 ¡ 4 m> 0

) ¡ 4 m>¡ 4

) m< 1

c For no real roots

¢< 0

) 4 ¡ 4 m< 0

) ¡ 4 m<¡ 4

) m> 1

C The discriminant of a quadratic

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Y:\HAESE\CAM4037\CamAdd_03\073CamAdd_03.cdr Thursday, 19 December 2013 3:18:36 PM GR8GREG