Cambridge International Mathematics

QUADRATIC EQUATIONS WITH NO REAL SOLUTIONS

Consider x^2 +2x+5=0.

Using the quadratic formula, the solutions are: x=

¡ 2 §

p

4 ¡4(1)(5)

2(1)

=

¡ 2 §

p

¡ 16

2

However, in the real number system,

p

¡ 16 does not exist. We

therefore say that x^2 +2x+5=0 has no real solutions.

If we graph y=x^2 +2x+5 we get:

The graph does not cut thex-axis, and this further justifies the

fact that x^2 +2x+5=0 has no real solutions.

We will discuss this more when we turn our attention to quadratic

functions.

EXERCISE 21C.2

1 Show that the following quadratic equations have no real solutions:

a x^2 ¡ 3 x+12=0 b x^2 +2x+4=0 c ¡ 2 x^2 +x¡1=0

2 Solve forx, where possible:

a x^2 ¡25 = 0 b x^2 +25=0 c x^2 ¡7=0

d x^2 +7=0 e 4 x^2 ¡9=0 f 4 x^2 +9=0

g x^2 ¡ 4 x+5=0 h x^2 ¡ 4 x¡5=0 i x^2 ¡ 10 x+29=0

j x^2 +6x+25=0 k 2 x^2 ¡ 6 x¡5=0 l 2 x^2 +x¡2=0

Aquadratic functionis a relationship between two variables which can be written in the form

y=ax^2 +bx+c wherexandyare the variables anda,b, andcare constants, a 6 =0.

Using function notation, y=ax^2 +bx+c can be written as f(x)=ax^2 +bx+c.

FINDINGyGIVENx

For any value ofx, the corresponding value ofycan be found by substitution into the function equation.

D QUADRATIC FUNCTIONS [3.2]

y

x

(-1' 4)

-6 -4 -2 2 4

15

10

5

O

Quadratic equations and functions (Chapter 21) 429

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Y:\HAESE\IGCSE01\IG01_21\429IGCSE01_21.CDR Tuesday, 18 November 2008 12:02:56 PM PETER