Solving Equations Worksheet1 Solution6





In this page solving equations worksheet1 solution6 we are going to see solution of practice questions.

Question 6:

Solving equation z² - 6 z + 9 = 4 √(z² - 6 z + 6) following roots are obtained

Solution:

z² - 6 z + 9 = 4 √(z² - 6 z + 6)

To remove square root on right side we have to take squares on both sides.

(z² - 6 z + 9)² = [4 √(z² - 6 z + 6)]²

(a - b + c)² = a² + b² + c² - 2 a b - 2 b c + 2 c a

(z² - 6 z + 9)² = (z²)² + (6z)² + (9)² - 2 (z²)(6z) - 2(6z) (9) + 2 (9) (z²)

                   = z⁴ + 36 z² + 81 - 12 z³ - 108z + 18 z²

(z² - 6 z + 9)² = [4 √(z² - 6 z + 6)]²

[z⁴ + 36 z² + 81 - 12 z³ - 108z + 18 z²] = 16 (z² - 6 z + 6)

[z⁴ + 36 z² + 81 - 12 z³ - 108z + 18 z²] = 16z² - 96 z + 96

z⁴ - 12 z³ + 36 z² + 18 z² - 16z² - 108z + 96 z + 81 - 96 = 0

z⁴ - 12 z³ + 38 z² - 12 z - 15 = 0


from the synthetic division we have found two two roots those are (z - 1) (z - 5)

By solving the remaining quadratic equation we can find other two roots

z² - 6 z - 3 = 0

we cannot factorize this quadratic equation,so let us use formula to solve this equation.

  z = [- b ± √(b² - 4 ac)]/2a

a = 1  b = -6 and c = -3

  z = [-(-6) ± √(-6)² - 4 (1)(-3))]/2(1)

  z = [6 ± √(36 + 12)]/2

  z = [6 ± √48]/2

  z = [6 ± 4 √3]/2

  z = 2[3 ± 2 √3]/2

  z = 2[3 + 2 √3]/2                       z = 2[3 - 2 √3]/2

  z = 3 + 2 √3                               z = 3 - 2 √3

solution are 1,5,3 + 2 √3,3 - 2 √3

Therefore correct option is option (C)

(A)  3 + 2 √3 , 3 - 2 √3

(B)  5 , 1

(C)  all the above

(D)  none of the above

solving equations worksheet1 solution6 solving equations worksheet1 solution6