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