Question

Find the real solutions of the quadratic equation $\frac{1}{x-1}+\frac{1}{x+2}=\frac{5}{4}$.

Short answer

The given quadratic has two real solutions $x=2,-\frac{7}{5}$.

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$\frac{1}{x-1}+\frac{1}{x+2}=\frac{5}{4}$ which can be written as,

role="math" localid="1643831365940" $$\begin{gathered} \frac{1}{x-1}+\frac{1}{x+2}=\frac{5}{4}\mathrm{....}\left(\text{Given}\right) \\ \text{or,\hspace{0.17em}}\frac{x+2+x-1}{\left(x-1\right)\left(x+2\right)}=\frac{5}{4} \\ \frac{2x+1}{x^2+x-2}=\frac{5}{4} \\ 5x^2+5x-10=8x+4....\left(\text{Crossmultiply}\right) \\ 5x^2-3x-14=0....\left(1\right) \end{gathered}$$

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

$$\begin{gathered} x=\frac{-\left(-3\right)\pm \sqrt{{\left(-3\right)}^2-4\left(5\right)\left(-14\right)}}{2\left(5\right)} \\ =\frac{3\pm \sqrt{289}}{10} \\ =\frac{3\pm 17}{10} \\ =2,-\frac{7}{5} \end{gathered}$$

Step 3. Description of steps.

The given quadratic equation has two distinct real solutions $x=2,-\frac{7}{5}$.