SOLVE THE  EQUATION BY FINDING SQUARE ROOTS

Problem 2 :

x² - 6x + 9 = 25

Solution :

x² - 6x + 9 = 25

x² - 6x + 9 - 25 = 0

x² - 6x - 16 = 0

Write the trinomial on the left side of the above equation in terms of square of a binomial.

x² - 2(x)(3) - 16 = 0

x² - 2(x)(3) + 3² - 3² - 16 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 3)² - 3² - 16 = 0

(x - 3)² - 9 - 16 = 0

(x - 3)² - 25 = 0

(x - 3)² = 25

Take square root on both sides.

√ (x - 3)² = √25

x - 3 = ±5

x - 3 = 5      x - 3 = -5

x = 8          x = -2

Therefore, the solutions are 

x = 8, x = -2

Problem 3 :

x² - 12x + 36 = 49

Solution :

x² - 12x + 36 = 49

x² - 12x + 36 - 49 = 0

x² - 12x - 13 = 0

Write the trinomial on the left side of the above equation in terms of square of a binomial.

x² - 2(x)(6) - 13 = 0

x² - 2(x)(6) + 6² - 6² - 13 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 6)² - 6² - 13 = 0

(x - 6)² - 36 - 13 = 0

(x - 6)² - 49 = 0

(x - 6)² = 49

Take square root on both sides.

√ (x - 6)² = √49

x - 6 = ±7

x - 6 = 7      x - 6 = -7

x = 13          x = -1

Therefore, the solutions are 

x = 13, x = -1

Problem 4 :

2x² - 12x + 18 = 32

Solution :

2x² - 12x + 18 = 32

2x² - 12x + 18 - 32 = 0

2x² - 12x - 14 = 0

2(x² - 6x - 7) = 0

x² - 6x - 7 = 0

Write the trinomial on the left side of the above equation in terms of square of a binomial.

x² - 2(x)(3) - 7 = 0

x² - 2(x)(3) + 3² - 3² - 7 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 3)² - 3² - 7 = 0

(x - 3)² - 9 - 7 = 0

(x - 3)² - 16 = 0

(x - 3)² = 16

Take square root on both sides.

√(x - 3)² = √16

x - 3 = ±4

x - 3 = 4      x - 3 = -4

x = 7          x = -1

Therefore, the solutions are 

x = 7, x = -1

Problem 5 :

4x² - 4x + 1 = 36

Solution :

4x² - 4x + 1 = 36

4x² - 4x + 1 - 36 = 0

4x² - 4x - 35 = 0

x² - x - 35/4 = 0

Write the trinomial on the left side of the above equation in terms of square of a binomial.

x² - 2(x)(1/2) - 35/4 = 0

x² - 2(x)(1/2) + (1/2)² - (1/2)² - 35/4 = 0

Using the identity (a - b)² = a² - 2ab + b²

(x - 1/2)² - (1/2)² - 35/4 = 0

(x - 1/2)² - 1/4 - 35/4 = 0

(x - 1/2)² - 36/4 = 0

(x - 1/2)² = 36/4

(x - 1/2)² = 9

Take square root on both sides.

√(x - 1/2)² = √9

x - 1/2 = ±3

x - 1/2 = 3      x - 1/2 = -3

x = 7/2          x = -5/2

Therefore, the solutions are 

x = 7/2          x = -5/2