Solving Quadratic Equations    

A Quadratic equation is an equation in the form “ax² + bx + c = 0″, where x is variable, a, b, c, are real numbers, and a ≠ 0.

Example- (1) x² – 4x + 5 = 0

               (2) x²  + 3x – 4 = 0

               (3) 3x²  + 5x – 7 = 0 are Quadratic equations.

If p(x) is a polynomial of degree 2, of any equation in the form p(x) = 0, is a Quadratic equation.

But when we write the terms of polynomial p(x) in descending order of their degree, then we get the standard form of a Quadratic equation.
that is “ax² + bx + c = 0″, a ≠ 0.

There are 3 methods for solving Quadratic equations.

(1) Factorisation

(2) Method of completing the square 

(3) Using the quadratic formula

Below are steps to solve the quadratic equation by factorisation.

                        Factorisation

(1) First we put all terms of the equation on one side, of equal sign and other side leaving zero.

(2) Factor the non zero side.

(3) Set each factor equal to zero.

(4) Solve each of these equation.

(5) Check answer by put value in original equation.

Example- (1)  Solve the equation  x²  – 3x = 10 

(1) First we put all terms of the equation on one side, of equal sign and other side leaving zero.

x²  – 3x – 10 = 0

(2) Factor the non zero side.

                     x²  – 5x + 2x – 10 = 0

                          (x – 5) (x + 2) = 0

(3) Set each factor equal to zero.

            (x – 5) = 0   or    (x + 2) = 0

(4) Solve each of these equation.

                    x = 5   or   x = -2 

(5) Check answer by put value in original equation.

x²  – 3x = 10                         

put x = 5 

                 5²  – 3 x 5 = 10 

                          25 – 15 = 10                         

10 = 10 

                                           or 

                    x²  – 3x = 10                     

  put x = -2   

            (-2)² – 3 x (-2) = 10  

            4 + 6 = 10     

            10 = 10         

The solution is x = {5 , -2}

Example- (2) Solve the equation  x² – 10x  = -16

 (1) First we put all terms of the equation on one side, of equal sign and other side leaving zero. 

                         x²  – 10x + 16 = 0

(2) Factor the non zero side. 

x²  – 8x – 2x + 16 = 0

(x – 8) (x – 2) = 0

(3) Set each factor equal to zero.

(x – 8) = 0   or    (x – 2) = 0

(4) Solve each of these equation.

                     x = 8   or   x = 2 

(5) Check answer by put value in original equation.

x²  – 10x = -16

                           put x = 8

                   8²  – 10 x 8 = -16 

                      64 – 80 = -16

                             -16 = -16 

                                     or

                 x²  – 10x = -16

put x = 2  

               (2)² – 10 x (2) = -16  

           4 – 20  = -16

        -16 = -16

             The solution is x = {8 , 2}

Example- (3) Solve the equation  x²  – 6x = -8

 (1) First we put all terms of the equation on one side, of equal sign and other side leaving zero. 

                         x²  – 6x + 8 = 0

(2) Factor the non zero side. 

x²  – 4x – 2x – 8 = 0

(x – 4) (x – 2) = 0

(3) Set each factor equal to zero.

(x – 4) = 0   or   (x – 2) = 0

(4) Solve each of these equation.

x = 4   or   x = 2 

(5) Check answer by put value in original equation.

x²  – 6x = -8 

                         put x = 4                   

4² – 6 x 4 = -8 

                                      16 – 24 = -8 

                           -8 = -8 

                                   or 

                        put x = 2 

                     x²  – 6x = -8  

            (2)² – 6 x (2) = -8  

                        4 – 12 = -8

                              -8 = -8

The solutions are x = 3  and  x = 2

                         Common Factors

Example- (1)   What are the factors of 9x² – 3x = 0

Here 3 is a common factor of 9 and 3,

Here we see that 3 is a common in both terms.

We take out 3 from each term.

so, 3(3x²  – x) = 0

x²  and x also common factor of x

3x(3x- 1) = 0

so, the factors are 3x and (3x – 1)

Set each factor equal to zero

either 3x = 0  or   (3x – 1) = 0

          Solve each of these equation

x = 0   or   3x = 1

                 x = 0  or  x = 1/3

      The solutions are x = 0 and x = 1/3

Example- (2)   What are the factors of x² – 5x = 0

Here we see that x is a common in both terms.

We take out x from each term.
                    so, x(x – 5) = 0

Here x is a common factor of x² and x,

x(x- 5) = 0

so, the factors are x and (x – 5)

Set each factor equal to zero

either x = 0  or   (x – 5) = 0

          Solve each of these equation.

                    x = 0   or   x = 5

     The solutions are x = 0 and x = 5

Example- (3)   What are the factors of 5x² = 30x

Rewrite the equation as transform 30x from right side to left side.
                       5x²  – 30x = 0

Here we see that 5 is a common in both terms.We take out 5 from each term.

so, 5(x²  – 6x) = 0

Here x is a common factor of x² and 6x,

                        5x(x- 6) = 0

so, the factors are 5x and (x – 6)

Set each factor equal to zero.

either 5x = 0  or   (x – 6) = 0

          Solve each of these equation.

                    x = 0   or   x = 6

     The solutions are x = 0 and x = 6

(2) Method of completing the square

 Example- (1) Solve the equation 

                        x²  + 4x  + 4 =  9

              We can write the equation as

                    x²  + 2x + 2x + 4 = 9 

                   x(x +2) + 2(x + 2) = 9

                        (x + 2)(x + 2) = 9

                          (x + 2)²  = 9 

                          (x + 2)²  = ±3² 

                            (x + 2) = ±3

          (x + 2) = +3     or       (x + 2) = -3

               x = +3 – 2    or     x = -3 – 2

                         x = 1  or   x = -5

               The solutions are 1 and  -5