Cm qe
1
The general form of a quadratic equation is ax 2 + 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 ( 29( 29 β α β α β α = = = - = - = - - x x x x x x or or 0 0 0 b) By completing the square eg: 2x 2 - 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 ac b b x 2 4 2 - ± - = 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 x 2 - ( 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: x 2 – ( sum of the roots ) x + product of the roots = 0 3) The Quadratic Equation ax 2 + bx + c = 0 can thus be expressed as x 2 - ( 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 b 2 - 4ac Determinant The Types of Roots b 2 - 4ac > 0 Two different roots (two distinct roots) b 2 - 4ac = 0 Two equal roots (one root) b 2 - 4ac < 0 No real root (no root) SAMPLE QUESTIONS Find the range of x if the straight line y = 2x + k a) intersects the curve x 2 + y 2 – 6 = 0 at two different points. b) is the tangent of the curve x 2 + y 2 – 6 = 0. c) does not intersect the curve x 2 + y 2 – 6 = 0. Use b 2 - 4ac > 0 Use b 2 - 4ac = 0 Use b 2 - 4ac < 0
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Transcript of 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
