Question 1086139: Suppose a and x satisfy x^2 + (a-(1/a))x - 1 = 0. Solve for x in terms of a.
*That's a quadratic in x Found 2 solutions by Fombitz, ikleyn:Answer by Fombitz(32388) (Show Source):
You can put this solution on YOUR website! Use a substitution for ease,
Complete the square,
So then,
and
So then substituting,
"Positive" solution:
"Negative" solution:
I got two solutions. Each is as short as 2 - 4 lines.
Solution 1 (the Vieta's theorem; 4-lines solution)
The Vieta's theorem says: if p and q are the roots of a quadratic equation = 0 then u = -(p+q) and v = pq.
The opposite is also TRUE: if u = -(p+q) and v = pq then p and q are the roots of the quadratic equation = 0.
Now look into your equation and notice that the numbers -a and give when summed up and -1 when multiplied.
Hence, the numbers "-a" and are the roots of your equation.
Solution 2 (Factoring. 2-lines solution)
Factor your polynomial: = .
Hence, the roots are "-a" and .
