CONTENT LIST
QUADRACTIC EQUATION
Introduction
The polynomial equation whose highest degree is two is
called a quadratic equation.
General form of quadratic polynomial is ax2 + bx
+ c, a ≠ 0. When we equate this polynomial to zero, we get a quadratic
equation.
Thus, quadratic equation is ax2 + bx + c= 0 where a, b and c being constants
(a ≠ 0).
Example:
x2 +
4x + 4 = 0 (standard form of quadratic equation)
2x2 – 64= 0 (Missing linear coefficient)
x2 -7x =0
(Missing constant term)
(x + 2)(x - 3) = 0 (Quadratic equation
example in factor form)
Quadratic equation have two real Solutions.
An
interesting fact about quadratic equations is that it can have up to two real
solutions. Solutions are values of variable where the quadratic equals 0. Real
solutions mean that solutions are not imaginary and they are real numbers.
Solution of Quadratic equation
The solutions of a quadratic equation ax2 + bx + c = 0 Where
a, b and
c being constants (a ≠ 0) are given by the quadratic formula:
x = [-b ± √(b² -
4ac)] / (2a)
Here (b² - 4ac) is called discriminant of quadratic
equation. That is denoted by D.
D= (b² - 4ac)
Discriminant is used to determine the nature of roots of
the quadratic equation.
Nature of roots of the quadratic equation
·
If D > 0, then the equation ax2 + bx + c
= 0 has two real and
distinct solutions (roots).
·
If D = 0, then the equation ax2 + bx + c =
0 has only one real solution (root)or equal roots.
·
If D < 0, then the equation ax2 + bx + c
= 0 has two distinct complex roots.
By using the
discriminant, we can determine the number of solutions of quadratic equations
without actually solving it.
Sum and product of roots
For any quadratic equation ax2 +
bx + c = 0, where a, b and c being constants (a ≠ 0).
·
The sum of roots = -b/a
·
The product of the roots = c/a.
Graph of the quadratic equation
·
The graph of a quadratic function is a U-shaped
curve termed a parabola.
·
The sign on the coefficient of “a” the
quadratic function dictates whether the graph opens up or down.
·
If a<0, the graph opens down and
if a>0 then the graph opens up.
Solution of quadratic equation
Method 1: Factoring method
To solve a quadratic equation by factoring method,
1. Put all terms on LHS
side of the equal sign zero on the RHS side.
2. Factorize LHS side
polynomial.
3. Set each factor equal to
zero.
4. Find value of variable
by equating each factor with zero.
5. Check by inserting your solutions
in the original equation.
Example: Solve x 2 –
6 x = 16.
Step1: Put all terms
on LHS side of the equal sign zero on the RHS side
x 2 –
6 x = 16
=> x 2 – 6 x – 16 =
0
Step 2: Factorize LHS
side polynomial.
=> ( x – 8)( x + 2) = 0
Step 3: Set each
factor equal to zero.
( x – 8)=0
=> x=8
( x + 2) =0
=> x=-2
Step5: Check by inserting your solutions in the original equation.
When x=8,
LHS=(8)2- 6*8 =64-48=16 Here ,LHS =RHS
When x=-2,
LHS=(-2)2- 6*(-2) =4+12=16 Here ,LHS =RHS
Thus, roots of x 2 –
6 x = 16 are 8,-2.
Method 2: Using standard formula for roots of quadratic equation
Many quadratic equations
cannot be solved by factoring method. This is required when the roots are not
rational numbers.
A second method of solving quadratic equations needs the use of
the following formula:
a, b, and c are taken from the quadratic equation
by comparing with general form of quadratic equation (ax 2 + bx + c =
0)
Example: Solve x 2 –
6 x = 16.
Given quadratic equation is x 2 –
6 x = 16 or x 2 – 6 x –
16=0
Now comparing with standard form ax 2 + bx + c =
0 we get
a=1, b=(-6) and c=(-16)
using formula for x,
X=[ -(-6) ± √({-6}² - 4*1*{-16}]/2*1
X=(6±10)/2
When we take x=(6+10)/2=>x=8
When we take x=(6-10)/2=>x=-2
