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Page 1: Cm qe

The general form of a quadratic equation is ax2 + b x + c = 0,where a ,b, and c are constants and a ≠ 0 .The highest power of the variable, x , is 2

QUADRATIC EQUATIONS

HOW TO DETERMINE THE ROOTS

OF THE QUADRATIC EQUATION

a) By factorization

( ) ( )

βαβα

βα

===−=−

=−−

xx

xx

xx

oror 00

0

b) By completing the square

eg: 2x2 - 8x+5 = 2(x-2)2 – 3 = 0

( x - 2) 2 = 2

3

(x -2) = ± 1.2247

x = 3.2247,

or x = 0.7753

c) By using the formula

a

acbbx

2

42 −±−=

d) By using calculator

The root of a quadratic equation is the value that can replace the variable in the eguation to satisfies the equation.

HOW TO DETERMINE

WHETHER A GIVEN

VALUE IS THE ROOT OF

THE QUADRATIC

EQUATION

a) by substitution

b) by inspection

The roots of quadratic

equations can be

determined by trial

and improvement

method.

(ie the repeated

substitution of

integers into a

function or polynomial

to find solutions)

HOW TO DETERMINE THE QUADRATIC

EQUATION GIVEN THE ROOTS

1. If α and β are the roots of the Q.E

the equation is (x - α )(x - β ) = 0

or x2 - ( α + β )x + (αβ ) = 0

2. The Step of forming a quadratic equation from given roots are

i. Find the sum of the roots ( α + β )

ii. Find the product of the roots (αβ )

iii. Form a quadratic equation by writing in the following form:

x2 – ( sum of the roots ) x + product of the roots = 0

3) The Quadratic Equation

ax2 + bx + c = 0

can thus be expressed as

x2 - ( S.O.R) x + (P.O.R) = 0

where

S.O.R = sum of the roots = a

b−

P.O.R = product of the roots = a

c

HOW TO DETERMINE THE TYPES OF

ROOTS OF QUADRATIC

EQUATIONS

Find the value of the determinant

b2 − 4ac

Determinant The Types of Roots

b2 - 4ac > 0 Two different roots

(two distinct roots)

b2 - 4ac = 0 Two equal roots

(one root)

b2 - 4ac < 0 No real root

(no root)

SAMPLE QUESTIONSFind the range of x if the straight line

y = 2x + k

a) intersects the curve x2 + y2 – 6 = 0 at two different points. b) is the tangent of the curve x2 + y2 – 6 = 0.

c) does not intersect the curve x2 + y2 – 6 = 0.

Use b2 - 4ac > 0

Useb2 - 4ac = 0

Useb2 - 4ac < 0