A polynomial of degree 2 is called Quadratic Equation.
For example:
- 2x2-5x+1=0
- x2-5=0
A general form of quadratic equation is ax2+bx+c=0
Where a, b, c all belong to real numbers
Now if you compare the equations I and II with the general form.
- a=2, b=-5 and c=1
- a=1, b=0 and c=-5
Zeros or the solutions of Quadratic Equation
The real value of x for which the value of the P(x) = ax2+bx+c becomes zero is known as the root of
the quadratic equation ax2+bx+c =0.
- Determine whether 3and 4 are the zeros of the polynomial P(x) =x2-7x+12
Solution:
P (3) = 32-7.3+12
= 9-21+12
= -12+12
= 0
P (4) = 42-7.4+12
= 16-28+12
= -12+12
= 0
Therefore 3 and 4 are the zeros or the solutions or the roots of the polynomial P(x)=x2-7x+12
Kinds of a Quadratic Equation
There are two types of quadratic equations
- Pure quadratic equation
- Adfected quadratic equation
Pure quadratic equation:
An equation of the form ax2+c=0 , a ≠0 is known as the pure quadratic equation .It means that the
quadratic equation ax2+bx+c =0, having no term containing single power of x, known as pure
quadratic equation . Clearly in a quadratic equation a ≠0 and b=0.
For example
- x2-4=0
- 3/8x2=5
Adfected quadratic equation:
A quadratic equation of the form ax2+bx+c =0, a ≠0 is known adfected quadratic equation or general
quadratic equation. An adfected quadratic equation has also a term containing single power of x .In
adfected quadratic equation
ax2+bx+c =0, a ≠0, b ≠0
For example:
- x2-5x+11=0
- 0.3x2+17x-2.3=0
Solving a pure quadratic equation
There are two methods for solving a pure quadratic equation of the form ax2+c=0:
- By square root
- By factorization
To solve a quadratic equation by square root
ax2+c=0 is a pure quadratic equation. To solve it , bring the constant term the RHS (right hand side)
and divide both side by a, coefficient of x2 and take the square root.
For example :
4x2-49=0
4x2=49
x2=49/4
x = +7/2
To solve a pure quadratic equation by factorization
Bring the equation ax2+c=0 in the form p2-q2 =0. Use the p2-q2=(p+q)(p-q). Equate each factor to
zero and find the values of x in each case, the two values of x so obtained are roots of the equation
ax2+c=0
For example :
16x2-25=0
(4x)2-(5)2 = 0
(4x+5) (4x-5) =0
4x-5 =0
x= 5/4
4x+5 =0
x =-5/4
Solving an Adfected Quadratic Equation
There are two methods for solving an adfected quadratic equation ax2 + bx+c=0, a ≠0, b ≠0.
(i) By factorization (ii) By completing square
To solve the quadratic equation ax2 + bx+c=0 by the method factorization
In this method the middle term (i.e term containing single power of x) is broken into two suitable
parts so that the factors are formed.
Example: solve. 8x2 + 7x – 15 = 0
Solution: 8x2 +7x- 15 =0
Or 8x2-8x + 15x-15=0
Or 8x (x-1) +15(x-1) =0
Or (8x +15)(x-1)=0
Therefore 8x+15=0
8x=-15
x= -15/8
And x-1 =0
x-1
Hence -15/8 and 1 are the required roots.
To solve ax2 + bx+ c=0 by completion of square
The famous Indian mathematician ShreedharAcharya had invented a formula for solving the
quadratic equation ax2 +bx+c=o.
If the equation ax2 +bx +c =0 has roots α and β, then
and
Where a = coefficient of x2
b = coefficient of x,
c = constant term
