How many extraneous solutions does the equation below have?
$\frac{2 m}{2 m+3}-\frac{2 m}{2 m-3}=1$
We are given the rational equation \(\frac{2m}{2m+3} - \frac{2m}{2m-3} = 1\). To solve this, we first identify the values of \(m\) that make the equation undefined. These occur when the denominators are zero: \(2m+3=0\) and \(2m-3=0\), which correspond to \(m = -\frac{3}{2}\) and \(m = \frac{3}{2}\), respectively. Any solution obtained that equals either \(\frac{3}{2}\) or \(-\frac{3}{2}\) would be an extraneous solution, as it would cause division by zero in the original equation.
To solve the equation, we eliminate the denominators by multiplying the entire equation by the common denominator, which is \((2m+3)(2m-3) = 4m^2 - 9\). The equation becomes:
\(\frac{2m(2m+3)(2m-3)}{2m+3} - \frac{2m(2m+3)(2m-3)}{2m-3} = 1(4m^2 - 9)\).
This simplifies to:
\(2m(2m-3) - 2m(2m+3) = 4m^2 - 9\).
Next, we expand and simplify the expression:
\(4m^2 - 6m - (4m^2 + 6m) = 4m^2 - 9\)
\(4m^2 - 6m - 4m^2 - 6m = 4m^2 - 9\)
\(-12m = 4m^2 - 9\).
Rearranging this into a standard quadratic equation format \(am^2 + bm + c = 0\):
\(4m^2 + 12m - 9 = 0\).
We solve the quadratic equation \(4m^2 + 12m - 9 = 0\) using the quadratic formula \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\(m = \frac{-12 \pm \sqrt{12^2 - 4(4)(-9)}}{2(4)} = \frac{-12 \pm \sqrt{144 + 144}}{8} = \frac{-12 \pm \sqrt{288}}{8}\).
Simplifying \(\sqrt{288} = \sqrt{144 \times 2} = 12\sqrt{2}\):
\(m = \frac{-12 \pm 12\sqrt{2}}{8} = \frac{-3 \pm 3\sqrt{2}}{2}\).
Thus, the potential solutions are \(m_1 = \frac{-3 + 3\sqrt{2}}{2}\) and \(m_2 = \frac{-3 - 3\sqrt{2}}{2}\).
Finally, we check the potential solutions against our identified undefined values \(m = \pm \frac{3}{2}\). Since \(\frac{-3 \pm 3\sqrt{2}}{2} eq \pm \frac{3}{2}\), neither of the solutions causes a denominator to be zero. Therefore, there are no extraneous solutions. The final answer is \(0\).
Understanding extraneous solutions is vital in various fields like engineering and economics, where mathematical models often involve ratios. For instance, in electrical engineering, the impedance of certain circuits might be modeled by rational functions where specific component values (like resistance or frequency) cannot be chosen because they would lead to division by zero, creating a singularity in the circuit's response. Recognizing these values ensures that our physical models are well-behaved and that the mathematical solutions are physically meaningful.