Question

How do you solve the equation \[\dfrac{15}{x}-\dfrac{15}{x-2}=-2\]?

Answer 1

Hint: In this problem, we have to solve the given fraction and find the value of x. We can see that the given fraction does not have a similar denominator and so we cannot subtract it directly. We can now cross multiply the given fraction and simplify the terms in the left-hand side and the right-hand, we will get a quadratic equation, which we have to solve using the quadratic formula to get the value of x.

Complete step-by-step solution:
We know that the given fraction to be solved is,
\[\dfrac{15}{x}-\dfrac{15}{x-2}=-2\]
We can now see that the given fraction cannot be directly subtracted as it does not have a similar denominator.
We can now use the cross-multiplication method, we get
\[\Rightarrow \dfrac{15\left( x-2 \right)-15x}{x\left( x-2 \right)}=-2\]
We can now multiply the terms inside the bracket in both the numerator and the denominator, we get
\[\Rightarrow \dfrac{15x-30-15x}{{{x}^{2}}-2x}=-2\]
We can now multiply the term \[{{x}^{2}}-2x\] in the left-hand side and the right-hand side and we can cancel the similar terms in left-hand side, we get
 \[\Rightarrow -30=-2\left( {{x}^{2}}-2x \right)\]
We can now multiply the term inside the bracket in the right-hand side, we get
\[\Rightarrow 2{{x}^{2}}-4x-30=0\]
We can now solve the above quadratic equation using the quadratic formula,
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
Where, a = 2, b = -4, c = -30.
We can substitute the above values in the quadratic formula and simplify it.
\[\begin{align}
  & \Rightarrow x=\dfrac{4\pm \sqrt{16-4\left( 2 \right)\left( -30 \right)}}{4} \\
 & \Rightarrow x=\dfrac{4\pm \sqrt{16+240}}{4}=\dfrac{4\pm 16}{4} \\
\end{align}\]
We can now solve the above step, we get
\[\begin{align}
  & \Rightarrow x=\dfrac{4+16}{4}=5 \\
 & \Rightarrow x=\dfrac{4-16}{4}=-3 \\
\end{align}\]
Therefore, the value of x is 5 and -3.

Note: Students make mistakes while cross multiplying the term, where we should cross multiply the numerators and denominators respectively and the both denominators. We can check the answer by substituting it in the given equation.
We can now substitute x = 5 in given equation \[\dfrac{15}{x}-\dfrac{15}{x-2}=-2\], we get
\[\begin{align}
  & \Rightarrow \dfrac{15}{5}-\dfrac{15}{5-2}=-2 \\
 & \Rightarrow 3-5=-2 \\
 & \Rightarrow -2=-2 \\
\end{align}\]
We can now substitute x = -3 in \[\dfrac{15}{x}-\dfrac{15}{x-2}=-2\], we get
\[\begin{align}
  & \Rightarrow \dfrac{15}{-3}-\dfrac{15}{-3-2}=-2 \\
 & \Rightarrow -5+3=-2 \\
 & \Rightarrow -2=-2 \\
\end{align}\]
Therefore, the values are correct.