Quadratic Equations - How to Solve Quickly

Today I am sharing quick technique to solve Quadratic Equations quickly. I will also share practice questions on same to;

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