Solving Rational Equations That Lead to Quadratics
We’ve looked at how to solve rational equations before. Sometimes, when you’re working through a rational equation, you end up with a quadratic equation. If that happens, you can simplify and solve the quadratic using the methods we’ve learned, like factoring or using the quadratic formula. Keep in mind that sometimes, after solving, you might find that there are no valid solutions.
Recall an important concept in working with rational expressions: To add or subtract fractions, we first need to find a common denominator. This ensures that the fractions have the same base, allowing us to combine them effectively.
Let’s write the equation with this common denominator and simplify the terms:
[latex]\begin{align*} \text{Original equation} & : & \frac{-4x}{x-1} + \frac{4}{x+1} &= \frac{-8}{x^2 - 1} \\ \text{Recognize common denominator} & : & x^2 - 1 &= (x-1)(x+1) \\ \text{Rewrite with common denominator} & : & \\ \text{Multiply the first term by } \frac{x+1}{x+1} & : & \frac{-4x(x+1)}{x^2-1} &= \frac{-4x^2 - 4x}{x^2-1} \\ \text{Multiply the second term by } \frac{x-1}{x-1} & : & \frac{4(x-1)}{x^2-1} &= \frac{4x - 4}{x^2-1} \\ \text{Combine terms over common denominator} & : & \frac{-4x^2 - 4x + 4x - 4}{x^2-1} &= \frac{-8}{x^2-1} \\ \text{Simplify numerator} & : & \frac{-4x^2 - 4}{x^2-1} &= \frac{-8}{x^2-1} \\ \end{align*}[/latex]
Now that we have matching denominators, we equate the numerators:
[latex]\begin{align*} x-1 &= 0 & x+1 &= 0 \\ x &= 1 & x &= -1 \end{align*}[/latex]
Lastly, check for extraneous solution:
Remember: we must check for extraneous solutions because the original equation had denominators that could be undefined for certain values of [latex]x[/latex].
Both [latex]x=1[/latex] and [latex]x=-1[/latex] make the original denominators zero, so they are not valid solutions.
Therefore, this equation actually has no solution.