Use completing the square to solve each of the following quadratic equations.

Question

a)x²−6x+1=0;

b)2x²+6x+7=0;

c)2x²+6x+7=0

d)3x²−2x−1=0

Answer 1

a)

x²−6x + 1  =  0

The equation x²−6x + 1  =  0

Separate variables and constants aside by subtracting 1 to each side.

x^2 - 6x  =  -1

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x^2 - 6x + 3^2  =  -1 + 3^2

(x - 3)^2  =  -1 + 9

(x - 3)^2  =  8

Take square root on both sides.

(x - 3)   =   ± √8

(x - 3)   =   ± 2√2

x   =  3  ± 2√2

Solutions are x = 3  + 2√2 and 3  - 2√2

b)

The equation 2x² + 6x + 7  =  0

Separate variables and constants aside by subtracting 7 each side

2x² + 6x  =  - 7

Here x^2 coefficient is 2 , for perfect square make x^2 coefficient 1 by dividing each side by 2

x² + 3x  =  - 7/2

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x² + 3x + (3/2)^2  =  - 7/2 + (3/2)^2

(x - 3/2)^2  =  - 7/2 + 9/4

(x - 3/2)^2  =  (-14 + 9)/4

(x - 3/2)^2  =  -5/4

Take square root on both sides.

(x - 3/2)   =   ± √(-5/4)

(x - 3/2)   =   ± i(√5)/2

(x - 3/2)   =   ± i(√5)/2

x   =   3/2 ± i(√5)/2

x   =   3/2 + i(√5)/2        ;         x   =   3/2 - i(√5)/2

d)

The equation 3x² − 2x − 1  =  0

Separate variables and constants aside by adding 1 to each side.

3x² − 2x   =  1

Here x^2 coefficient is 3 , for perfect square make x^2 coefficient 1 by dividing each side by 3

x² − 2x  =  1/3

To change the expression into a perfect square trinomial add (half the x coefficient)² to each side of the expression.

x² − (2x)/3 + (1/3)^2   =  1/3 + (1/3)^2

(x - 1/3)^2   =  1/3 + 1/9

(x - 1/3)^2   =  (3 + 1)/9

(x - 1/3)^2  =  4/9

Take square root on both sides.

(x - 1/3)   =   ± √(4/9)

(x - 1/3)   =   ± 2/3

x   =   1/3 ± 2/3

x   =   1/3 + 2/3         ;          x   =   1/3 - 2/3

x  =  1                       ;          x  =  -1/3

Answer :

a)  Solutions are x = 3  + 2√2 and x = 3  - 2√2

b)  Solutions are x   =   3/2 + i(√5)/2 and  x   =   3/2 - i(√5)/2

c)  Solutions are x   = 1  and  x = - 1/3.