typed Solve each equation 9x 3Squareroot x 2 x4 5x2 4 0

typed

Solve each equation. 9x + 3Squareroot x = 2 x^4 - 5x^2 + 4 = 0 x^2/3 - 4x^1/3 = -3

Solution

a) 9x +3sqrt(x) =2 ------------------------(1)

take 3sqrt(x) = u => 9x = u²

hence u² + u =2

=>   u² + u -2 =0

=> u² + (2-1)u -2 =0    => (u+2)(u-1) = 0

=> u =-2 and u =1   => u² =4 and u² = 1

=> 9x = 4 and 9x =1 => x =4/9 and 1/9   but (1) is satisfied only by x=1/9, so it is the final answer

b) x⁴ -5x² +4 = 0    --------------------------------(2)

put u = x² => u² = x⁴

u² -5u +4 = 0

=> u² -(4+1)u +4 =0 => (u-4)(u-1) = 0

=> u =4 and u =1

=> x = +-2   and x= +-1

all values found above are satisfying (2) and hence x = -2, -1, 1 and 2

C) x^2/3 -4 x^1/3 = -3             -------------------------(3)

put u = x^1/3    then u² =x^2/3

u² -4u = -3

=> u² -4u + 3 =0

=> u² -(3+1)u + 3 =0

=> (u-3)(u-1) = 0    => u =3 and u =1

=> x^1/3 = 3 and x^1/3 = 1

=> x = 9 and x =1

but only x =1 satisfy (3) , so it is the final solution