1. QUADRATIC EQUATIONS
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
The root of a quadratic equation is the value that can replace the variable in the eguation to satisfies the equation.
HOW TO DETERMINE THE TYPES OF
HOW TO DETERMINE HOW TO DETERMINE THE ROOTS HOW TO DETERMINE THE QUADRATIC ROOTS OF QUADRATIC
WHETHER A GIVEN OF THE QUADRATIC EQUATION EQUATION GIVEN THE ROOTS EQUATIONS
1. If α and β are the roots of the Q.E
VALUE IS THE ROOT OF
a) By factorization Find the value of the determinant
THE QUADRATIC
EQUATION ( x − α) ( x − β) = 0 the equation is (x - α )(x - β ) = 0 b2 − 4ac
x − α = 0 or x − β = 0 or x2 - ( α + β )x + ( αβ ) = 0 Determinant The Types of Roots
a) by substitution x =α or x=β
2. The Step of forming a quadratic b2 - 4ac > 0 Two different roots
b) by inspection equation from given roots are
(two distinct roots)
The roots of quadratic b) By completing the square i. Find the sum of the roots ( α + β ) b2 - 4ac = 0 Two equal roots
equations can be ii. Find the product of the roots ( αβ )
eg: 2x2 - 8x+5 = 2(x-2)2 – 3 = 0
iii. Form a quadratic equation by (one root)
determined by trial
3 writing in the following form:
and improvement ( x - 2) 2 = b2 - 4ac < 0 No real root
method.
2
x2 – ( sum of the roots ) x + product of the roots = 0
(no root)
(x -2) = ± 1.2247
(ie the repeated SAMPLE QUESTIONS
x = 3.2247,
substitution of 3) The Quadratic Equation Find the range of x if the straight line
integers into a or x = 0.7753 y = 2x + k
ax2 + bx + c = 0
function or polynomial a) intersects the
to find solutions) can thus be expressed as curve x2 + y2 – 6 = 0 at Use
c) By using the formula two different points. b2 - 4ac > 0
x2 - ( S.O.R) x + (P.O.R) = 0
− b ± b − 4ac
2
where b) is the tangent of the
x= curve x2 + y2 – 6 = 0.
Use
2a −b
b2 - 4ac = 0
S.O.R = sum of the roots =
a c) does not intersect the Use
d) By using calculator
curve x2 + y2 – 6 = 0. b2 - 4ac < 0
c
P.O.R = product of the roots =
a