Question

what positive number satisfies the condition that twice the number minus three times the reciprocal of the number is equal to 1?

Answer 1

Solve:

"twice the number minus three times the reciprocal of the number is equal to 1."
                                                     3(1)
Let the number be n.  Then 2n - ------- = 1
                                                       n

Mult all 3 terms by n to elim. the fractions:

2n^2 - 3 = n.  Rearranging this, we get 2n^2 - n - 3 = 0.

We need to find the roots (zeros or solutions) of this quadratic equation.

Here a=2, b= -1 and c= -3.  Let's find the discriminant b^2-4ac first:

disc. = (-1)^2 - 4(2)(-3) = 1 + 24 = 25.

That's good, because 25 is a perfect square.
                -(-1) plus or minus 5         1 plus or minus 5
Then x = ------------------------------ = --------------------------
                            2(2)                                  4

x could be 6/4 = 3/2, or -5/4.

You must check both answers in the original equation.  If the equation is true for one or the other or for both, then you have found one or more solutions.