Solving Quadratic Equation by Completing Square
Method of completing the square
Below are steps to solve the quadratic equation by Completing Square.
(1) First we set given equation 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 + 25 = 0
We can write the equation as
x2 + 5x + 5x + 25 = 0
x(x +5) + 5(x + 5) = 0
(x + 5)(x + 5) = 0
(x + 5)2 = 0
(x + 5) = 0
x = -5
The solutions is -5
Example- (3) 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
Example: (4) Solve the equation 36x2 – 48x + 25 = 9
We can write the equation as
36x2 – 48x + 25 – 9 = 0
36x2 – 48x + 16 = 0
(6x – 4)2 = 0
(6x – 4)=0
6x = 4
x = 4/6
x = 2/3
The solutions is 2/3
Example- (4) Solve the equation 5x2 – 6x -2 = 0
In above equation 5x2 is not a perfect square so, we multiply the equation by 5, we get
25x2 – 30x – 10 = 0
(5x)2 – 2(5x)3 + 32 – 32 – 10 = 0
(5x – 3)2 – 9 – 10 = 0
(5x – 3)2 – 19 = 0
(5x – 3)2 = 19
(5x – 3) = ± √19
5x = 3 ± √19
5x = 3 + √19
5x = 3 – √19
The solutions are x = (3 + √19)/5 and x = (3 – √19)/5