Help explain these quadratic(?) equations please

the title above the equations says: Quadratic(ish) Equations and Non-linear Inequalities
Its on a review for the test I'll be taking later but I still can't seem to figure these out. Despite all my studying math still proves to be my most difficult subject :(
can someone please help explain them, walk through them

(x+54)squared - (x+54) - 24 = 0

x - 3 = 10/x

x - 2 (√x) - 24 = 0

(x - 5)/(x+1) less than or = to 0

please and thank you. you don't have to do all of them just the ones you want

(x + 54)² - (x + 54) - 24 = 0
Let y = x + 54:
y² - y - 24 = 0
Use the quadratic formula:
y = (1 ± √[(-1)² - 4(1)(-24)]) / (2*1)
y = (1 ± √97) / 2
Substitute x+54 back for y:
x + 54 = (1 ± √97) / 2
x = (1±√97)/2 - 54
x = (1±√97)/2 - 108/2
x = (-108+1±√97)/2
x = (-107±√97)/2

x - 3 = 10/x
x(x - 3) = 10
x² - 3x = 10
x² - 3x - 10 = 0
(x - 5)(x + 2) = 0
x - 5 = 0, x + 2 = 0
x = 5, x = -2

x - 2√(x) - 24 = 0
Let y = √x. Then y² = x. So, substitute y for √x and y² for x:
y² - 2y - 24 = 0
(y - 6)(y + 4) = 0
y - 6 = 0, y + 4 = 0
y = 6, y = -4
Substitute √x for y:
√x = 6, √x = -4
We can't have √x = -4, because square roots can't be negative. So, we have:
√x = 6
x = 36

(x - 5)/(x + 1) ≤ 0
The denominator can't be 0, so we know x+1 isn't 0, meaning x isn't -1. Moreover, since the quotient of x-5 and x+1 is less than or equal to 0, we know x-5 and x+1 can't have the same sign, so we have:
x - 5 ≤ 0 and x + 1 ≥ 0, or x - 5 ≥ 0 and x + 1 ≤ 0
x ≤ 5 and x ≥ -1, or x ≥ 5 and x ≤ -1
The second option isn't possible, since anything greater than 5 is greater than -1. So, we are left with:
x ≤ 5 and x ≥ -1
-1 ≤ x ≤ 5
But we found in the beginning that x can't be -1, so the answer is:
-1 < x ≤ 5

Hope that helps :)

1) (x+54)² - (x+54) - 24 = 0 Think of the parentheses quantity as a single variable,