Find all real solutions 3t squareroot 69t 2SolutionGiven t

Find all real solutions. 3t + squareroot 6-9t = 2

Solution

Given that

3t + (6 - 9t) = 2

(6 - 9t) = 2 - 3t

Squaring on both sides of above equation

    [ (6 - 9t) ]² = ( 2 - 3t )²

   6 - 9t = 2² - 2.2.3t + ( 3t )² [ since, ( a - b )² = a² - 2ab + b² ]

6 - 9t = 4 - 12t + 9t²

   9t² -12t + 4 + 9t - 6 = 0

   9t² - 3t - 2 = 0

This equation is in the form of quadratic equation is, ax² + bx + c = 0

x = [ -b ± (b² - 4ac) ] / 2a

Here , a = 9 , b = - 3 , c = - 2

t = [ -(-3) ± ((-3)² -4.(9).(-2) ) ] /2.9

= [ 3 ± 81 ] / 18

= [ 3 ± 9 ] / 18

t = [ 1 ± 3 ] / 6

Hence,

t = [ 1 + 3 ] / 6 , [ 1 - 3 ] / 6

t = 4/6 , -2/6

t = 2/3 , -1/3

Therefore,

The real solutions are t =  2/3 , -1/3