# How do you solve (1/x)+1/(x+3)=1/4 using the quadratic formula?

May 3, 2016

$\textcolor{b l u e}{x = 6.772 \text{ and} - 1.772}$

#### Explanation:

Consider the left side

Common denominator is $x \left(x + 3\right)$

So we have:

$\frac{\left(x + 3\right) + x}{x \left(x + 3\right)} = \frac{1}{4}$

$\frac{2 x + 3}{{x}^{2} + 3 x} = \frac{1}{4}$

Multiply both sides by $\left({x}^{2} + 3 x\right)$

$\left(2 x + 3\right) \times \frac{{x}^{2} + 3 x}{{x}^{2} + 3 x} = \frac{{x}^{2} + 3 x}{4}$

but $\frac{{x}^{2} + 3 x}{{x}^{2} + 3 x} = 1$

$\left(2 x + 3\right) = \frac{{x}^{2} + 3 x}{4}$

Multiply both sides by 4

$4 \left(2 x + 3\right) = {x}^{2} + 3 x$

$8 x + 12 = {x}^{2} + 3 x$

Subtract $8 x$ and 12 from both sides

${x}^{2} + 3 x - 8 x - 12 = 0$

$\textcolor{b r o w n}{{x}^{2} - 5 x - 12 = 0}$
$\textcolor{b r o w n}{\text{Now we can use the formula}}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Standard form $y = a {x}^{2} + b x + c$
where $a = 1 \text{ ; "b=-5" ; } c = - 12$

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

$x = \frac{5 \pm \sqrt{{\left(- 5\right)}^{2} - 4 \left(1\right) \left(- 12\right)}}{2 \left(1\right)}$

$x = \frac{5 \pm \sqrt{25 + 48}}{2}$

$x = \frac{5 \pm \sqrt{73}}{2} \text{ }$ Note that 73 is a prime number

$\textcolor{b l u e}{x = 6.772 \text{ and} - 1.772}$