a) sqrt(4x+7) – 4 = sqrt(4x-1)

Square both sides. Remember to FOIL out the left side.

4x+7 -8sqrt(4x+7) + 16 = 4x-1

Combine like terms.

4x+23 – 8sqrt(4x+7) = 4x – 1

Move the terms out side the square root to the other side.

-8sqrt(4x+7) = 24

Divide both sides by -8

sqrt(4x+7) = -3

The square root of a number is never negative, so you will get no solution.

b) sqrt(1 + sqrt(24-10x)) = sqrt(3x+5)

Square both sides

1 + sqrt(24-10x) = 3x + 5

Subtract 1 from each side

sqrt(24-10x) = 3x +4

Square both sides. Remember to FOIL the right side.

24 – 10x = 9x^2 + 24x + 16

Move everything to one side.

9x^2 + 34x – 8 = 0

Factor.

(x+4)(9x-2) = 0

x+4 = 0 or 9x-2 = 0

x = -4 or x = 2/9

Go back and check these solutions to make sure they both work.

I have not checked them, so one or neither of them could not be the solution.

c) 2^(2x) – 3(2^x) + 1 + 8 = 0

First combine the 1 and 8

2^(2x) – 3(2^x) + 9 =0

Make a substitution. Let y = 2^x, then y^2 = 2^(2x).

y^2 – 3y + 9 = 0

Now use the quadratic formula to find y. Remember to go back and find x, and also check your answers.

d) Make a substitution here too. Let y = x^(1/4), then y^2 = x^(1/2)

y^2 – 17y + 16 = 0

Factor

(y-16)(y-1) = 0

y = 16 or y = 1.

Remember that y = x^(1/4), so

16 = x^(1/4) and 1 = x^(1/4)

Raise each side to the 4th power to get your answer.

Then check them in the original problem.