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.

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> 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.       Log On


   

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

Let w=x^2 to get

w%5E2-9w%2B18=0


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


Starting with the general quadratic

aw%5E2%2Bbw%2Bc=0

the general solution using the quadratic equation is:

w+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve w%5E2-9%2Aw%2B18=0 ( notice a=1, b=-9, and c=18)




w+=+%28--9+%2B-+sqrt%28+%28-9%29%5E2-4%2A1%2A18+%29%29%2F%282%2A1%29 Plug in a=1, b=-9, and c=18



w+=+%289+%2B-+sqrt%28+%28-9%29%5E2-4%2A1%2A18+%29%29%2F%282%2A1%29 Negate -9 to get 9



w+=+%289+%2B-+sqrt%28+81-4%2A1%2A18+%29%29%2F%282%2A1%29 Square -9 to get 81 (note: remember when you square -9, you must square the negative as well. This is because %28-9%29%5E2=-9%2A-9=81.)



w+=+%289+%2B-+sqrt%28+81%2B-72+%29%29%2F%282%2A1%29 Multiply -4%2A18%2A1 to get -72



w+=+%289+%2B-+sqrt%28+9+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



w+=+%289+%2B-+3%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



w+=+%289+%2B-+3%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

w+=+%289+%2B+3%29%2F2 or w+=+%289+-+3%29%2F2

Lets look at the first part:

x=%289+%2B+3%29%2F2

w=12%2F2 Add the terms in the numerator
w=6 Divide

So one answer is
w=6



Now lets look at the second part:

x=%289+-+3%29%2F2

w=6%2F2 Subtract the terms in the numerator
w=3 Divide

So another answer is
w=3

So our solutions are:
w=6 or w=3


Remember, we let w=x%5E2 so

x%5E2=6 or x%5E2=3

So our answers in terms of x are


or



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


Remember, if the discriminant b%5E2-4ac is equal to zero, then the quadratic will have one real solution


k%5E2-4%284%29%281%29=0 Set the discriminant equal to zero


k%5E2-16=0 Multiply


k%5E2=16 Add 16 to both sides


Take the square root of both sides


Simplify

So the values of k are k=-4 or k=4