How to solve [x / (x+3)] + [5 / (x-7)] = [30 / (x^2 - 4x - 21)]?

a)

x/(x+3) + 5/(x-7) = 30/(x^2-4x-21)

x/(x+3) + 5/(x-7) = 30/(x+3)(x-7)

x(x-7) + 5(x+3) = 30 ................. this is where I think you made a mistake, once all terms had a common denominator on both sides, you should have been able to drop the denominator by multiplying all terms by the denominator

x^2-7x+5x+15=30

x^2-2x-15=0

(x+3)(x-5)=0

x=3 or x=5

we can eliminate 3 because the original equation has (x-3) in the denominator of the first expression and so cannot be valid

b)

let's set A = sqrt(5x-4)

A = x/2 + 2

multiply by 2

2A = x + 4

square both sides

4A^2 = (x+4)^2

4A^2 = x^2 + 8x + 16

substitute for A^2

4(5x - 4) = x^2 + 8x + 16

20x - 16 = x^2 + 8x + 16

x^2 - 12x + 32 = 0

(x-8)(x-4)=0

x = 8 or 4

c)

x^(1/2) + (x-20)^(1/2) = 10

x^(1/2) = 10 - (x-20)^(1/2)

x = (10 - (x-20)^(1/2))^2 ......... square both sides

x = (10 - (x-20)^(1/2))(10 - (x-20)^(1/2))

x = 100 - 20(x-20)^(1/2) + (x-20) ...... FOIL

x = 100 - 20(x-20)^(1/2) + x - 20

0 = 80 - 20(x-20)^(1/2)

0 = 4 - (x-20)^(1/2)

(x-20)^(1/2) = 4

x-20 = 16

x = 36

What did you get for d?

2.2565791296208276532708239958782

x = 5 only one root

(x^2 - 2x + 15) / [(x-7)(x+3)] = 30 / [(x-7)(x+3)]

then simplify