Find the solution set of negative 𝑥 over 63 plus one over 𝑥 is equal to negative two over 63 in the set of real numbers.
Our first step is to multiply each of our terms by the common denominator 63𝑥. We can do this as 𝑥 is nonzero since the function is not defined at zero. This gives us negative 𝑥 over 63 multiplied by 63𝑥 plus one over 𝑥 multiplied by 63𝑥 is equal to negative two over 63 multiplied by 63𝑥. In the first term, the 63’s cancel. This leaves us with negative 𝑥 squared. In the second term, the 𝑥’s cancel, leaving us with positive 63. Finally, on the right-hand side, the 63s cancel once again, leaving us with negative two 𝑥. We have the quadratic equation negative 𝑥 squared plus 63 is equal to negative two 𝑥.
We can multiply through by negative one, giving us 𝑥 squared minus 63 is equal to two 𝑥. Subtracting two 𝑥 from both sides gives us 𝑥 squared minus two 𝑥 minus 63 is equal to zero.
One way of solving this quadratic is by factoring into two sets of parentheses. As the coefficient of 𝑥 squared is equal to one, the first term in each of these will be 𝑥. We now need to find two integers that have a product of negative 63 and a sum of negative two. Nine multiplied by seven is 63. Therefore, negative nine multiplied by seven is negative 63. Negative nine plus seven is equal to negative two. 𝑥 squared minus two 𝑥 minus 63 can be factored to give us 𝑥 minus nine multiplied by 𝑥 plus seven.
As the product of our parentheses equals zero, either 𝑥 minus nine equals zero or 𝑥 plus seven equals zero. This gives us two solutions: 𝑥 equals nine or 𝑥 equals negative seven. The solution set of the equation negative 𝑥 over 63 plus one over 𝑥 is equal to negative two over 63 are negative seven and nine.