Math Equation

[Tavrobel]

Notation: x^1/2 is also the same as a square root. Do not be misled. It means the same thing as something being under a radical.

So ultimately, what we want to do here, is that we want to find a value for the variable x. Unfortunately, it's hidden underneath of a square root. Though that might normally be trivial, we can assume that we should not just plug stuff into the calculator and be done with it. This means we have to get through the algebra. We'll need our FOIL methods, and you'll see why soon enough.

Let's write our equation; I assume it looks like this, but if it's wrong, try to correct me:
(3x - 8)^1/2 - (x - 4)^1/2 = 2

There's no way to square one side without still getting a radical somewhere. So, bring one of the terms over to the other side; in this case, we'll just use the second term, to make things simple:
(3x - 8)^1/2 = 2 + (x - 4)^1/2

Square both sides:
((3x - 8)^1/2)² = (2 + (x - 4)^1/2)²

Simplify; we need our foiling skills here:
(3x - 8) = 4 + 2(x - 4)^1/2 + x - 4

Cancel out the 4; isolate the radical by bringing the 2 and the x to the other side
(3x - 8 - x)/2 = (x - 4)^1/2

Combine like terms (as in, the variable x) and then square again:
((2x - 8)/2)²=((x - 4)^1/2)²

Simplify the 2 in the denominator, and clean up:
(x - 4)² = (x - 4)

Expand our squared term, and bring the right side over:
x² - 8x + 16 - x + 4 = 0

Combine like terms:
x² - 9x + 20 = 0

This looks like a regular polynomial. Factors of 20 are 1, 20; 2, 10; 4, 5. One of those pairs adds up to nine. So, factor terms:
(x - 5)(x - 4) = 0

Solve for x so that the equation equals zero:
x = 4 or 5

Does this make sense? Yes. Substituting in 4 or 5 into our original equation would not get any non-real answers. Additionally, since our highest exponent became a term to the second power, we should reasonably obtain two answers. Both 4 and 5 are separate and distinct answers.

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This is another case of foiling gone horribly imbasauce. What else is there to do other than to factor stuff out?

Okay, write your givens:
x⁴ - 12x² - 85 = 0.

Factors of 85 are indeed, 17 and 5. However, that twelve there is negative, which is 17 - 5. The 85 is also negative, which means that both of the terms in the parenthesis will not necessarily be the same sign.
(x² - 17)(x² + 5) = 0

Here's where you went wrong: the solution to (x² + 5) = 0 is not +/- (5)^1/2.
x² = -5.

The solution is +/- (5i)^1/2. i, being the square root of -1, when squared, will give us a negative, which is what is needed to solve the problem. So, our new solutions are:
+5i^1/2
-5i^1/2
+17^1/2
-17^1/2

+/- (17)^1/2 remains unchanged. Your initial guess about the solution to the second equation was correct.

EDIT: Okay, so you changed the situation on me. In this case, your guess with +/- (17i)^1/2 is incorrect. Though +/- (5i)^1/2 is correct, that does not mean that it applied to the 17, which had a negative factor with your x². That makes it normal. The positive factor in front of your 5 with x², made it need i as part of your solution.