SOLUTION: I need help with these two prolems: Solve:x^4-9x^2+18=0 & Find k so that 4x^2-kx+1=0 has one rational solution.

Question 109416: I need help with these two prolems:
Solve:x^4-9x^2+18=0
&
Find k so that 4x^2-kx+1=0 has one rational solution.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
"Solve:x^4-9x^2+18=0"

Let w=x^2 to get

Let's use the quadratic formula to solve for w:

Starting with the general quadratic

the general solution using the quadratic equation is:

So lets solve ( notice , , and )

Plug in a=1, b=-9, and c=18

Negate -9 to get 9

Square -9 to get 81 (note: remember when you square -9, you must square the negative as well. This is because .)

Multiply to get

Combine like terms in the radicand (everything under the square root)

Simplify the square root (note: If you need help with simplifying the square root, check out this solver)

Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

or

Lets look at the first part:

Add the terms in the numerator
Divide

So one answer is

Now lets look at the second part:

Subtract the terms in the numerator
Divide

So another answer is

So our solutions are:
or

Remember, we let so

or

So our answers in terms of x are

"Find k so that 4x^2-kx+1=0 has one rational solution. "

Remember, if the discriminant is equal to zero, then the quadratic will have one real solution

Set the discriminant equal to zero

Multiply

Add 16 to both sides

Take the square root of both sides

Simplify

So the values of k are or
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