Question

Find the real solutions of the quadratic equation $\frac{1}{x-1}-\frac{2}{x^2}=0$.

Short answer

The given quadratic has no real solutions and complex solutions are $x=1+i,1-i$.

Step-by-step solution

Step 1. By method of quadratic formula.

The roots of the quadratic equation of the form $a{x}^2+bx+c=0$where $a\ne 0$can be evaluated by using the quadratic formula shown below.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Step 2. Find roots of quadratic equation by quadratic formula.

The given quadratic equation is$\frac{1}{x-1}-\frac{2}{x^2}=0$ which can be written as,

role="math" localid="1643829766525" $$\begin{gathered} \frac{1}{x-1}-\frac{2}{x^2}=0....\left(\text{Given}\right) \\ \text{or,\hspace{0.17em}}\frac{1}{x-1}=\frac{2}{x^2} \\ {x}^2=2x-2 \\ {x}^2-2x+2=0....\left(1\right) \end{gathered}$$

Evaluate roots of (1) by quadratic formula as shown below.

role="math" localid="1643829945472" $$\begin{gathered} x=\frac{-\left(-2\right)\pm \sqrt{{\left(-2\right)}^2-4\left(1\right)\left(2\right)}}{2\left(1\right)} \\ =\frac{2\pm \sqrt{-4}}{2} \\ =\frac{2\pm 2i}{2}\mathrm{....}\left(\sqrt{-1}=i\right) \\ =1+i,1-i \end{gathered}$$

Step 3. DESCRIPTION OF STEPS.

The given quadratic equation has no distinct real solutions and complex solutions are $x=1+i,1-i$.