Introduction to Quadratic equation

What is a quadratic equation?

A quadratic equation in variable X is an equation in which the greatest exponent of the variable is two.

For example:

                                3x²+4x+2=0        

General form of quadratic equation

                                ax²+bx+c=0

Where a, b, c, ε R and a ≠o or where a,b,c are real numbers and a≠o

 If a=o

Than bx +c=0 which is not quadratic equation .it becomes linear equation.

If b=0

Then ax2+c=0 which is also a quadratic equation. But this is called a pure quadratic equation.

Example:

                         2x²+5x+7=0   (general form quadratic equation)

                             3x²+2=0          (   pure quadratic equation)

                                     X+1=0           (linear equation)

How to solve a quadratic equation?

• Factorize
• Completing  the  square 
• Using quadratic formula
• Graphing

What is roots?

                The solution of an equation is also called its roots.

Roots

• One root will be reciprocal to the other. If a=c

     The coefficient of x2 is equal to the constant.

For example:

2x²+5x+2=0

2x²+4x+x+2=0

2x(x+2) +1(x+2) =0

(2x+1)(x+2)=0

2x+1=0    x+2=0

2x=-1       x=-2

X= -1/2    x=-2

• One root is zero if   c=0

For example:

2x²+4x=0

2x(x+2) =0

2x=0       x+2=0

X=0           x= -2

• The roots are equal in magnitude but opposite  in direction (sign) if   b=0

For example:

X²-4=0

X²= 4
square roots on both side

X=±2

X=2      x= -2

• If   a+b+c=0 

If sum of all coefficient is equal to zero. Then 1, c/a are the roots 

For example:

2x²-5x+3=0

2x²-2x-3x+3=0

2x(x-1)-3(x-1) =0

(2x-3)(X-1)=0

2x-3=0      x-1=0

2x=3          x=1

X= 3/2          x=1      

  X=1          x= 3/2

• If  a-b+c =0 than the roots are   -1  , -c/a

For example:

                                                     2x²+5x+3=0

                                                     2x²+2x+3x+3=0

                                                     2x(x+1) +3(x+1) =0

                                                     (2x+3)(x+1)=0

                                                     2x=­­­­­­­-3       x=-1

                                                     X=-3/2        x=-1

• if one root is  is p+iq  than the second roots will be p-iq

• if one root  is p+√q than second root p- √q

• The quadratic equation whose roots are reciprocal of the roots of    ax²+bx+c=o   is cx²+bx+a=0

Coefficient written is reverse order

For example:

3x²+7x+4=0

3x²+3x+4x+4=0

3x(x+1) +4(x+1) =0

(3x+4)(x+1)=0

3x+4=0     x+1=0

3x=-4        x=-1

X= -4/3         x=-1

For example:

4x²+7x+3=0

4x²+4x+3x+3=0

4x(x+1) +3(x+1) =0

(4x+3)(x+1)=0

4x+3=0       x+1=0

4x=-3           x=-1

X= -3/4            x=-1

• If a=1 and b,c ε z then the roots are rational numbers and must be integers.

For example:

X²+5x+6=0

X²+3x+2x+6=0

X(x+3) +2(x+3) =0

(x+2)(x+3) =0

X+2=0    x+3=0

X=-2       x=-3

• ax² +bx +c=0 

Sum of the roots =S= -b/a

        b=coefficient of x

         a=coefficient of x²

Product of the roots =p=

C=constant

a=coefficient of x²

Frequently Asked Questions-FAQs