# How do you solve the equation: x + 1/x+1 = 4?

Jul 23, 2015

$x = \setminus \frac{3 \setminus \pm \sqrt{5}}{2} \setminus \approx 2.618 , 0.382$

#### Explanation:

Two methods:

1) Multiply everything by $x$ first to get ${x}^{2} + 1 + x = 4 x$, or ${x}^{2} - 3 x + 1 = 0$. The quadratic formula now gives $x = \setminus \frac{3 \setminus \pm \sqrt{9 - 4}}{2} = \setminus \frac{3 \setminus \pm \sqrt{5}}{2} \setminus \approx 2.618 , 0.382$.

2) Add the fractions on the left first by getting a common denominator of $x$:

${x}^{2} / x + \frac{1}{x} + \frac{x}{x} = 4 \setminus R i g h t a r r o w \frac{{x}^{2} + x + 1}{x} = \frac{4}{1}$

Now "cross-multiply" to get ${x}^{2} + x + 1 = 4 x$, or ${x}^{2} - 3 x + 1 = 0$. This gives the same answer now as Method 1: $x = \setminus \frac{3 \setminus \pm \sqrt{9 - 4}}{2} = \setminus \frac{3 \setminus \pm \sqrt{5}}{2} \setminus \approx 2.618 , 0.382$.

As with all algebra problems like this, you should check the answers in the original equation. If you try this with, for example, $x = \frac{3 + \sqrt{5}}{2}$ in exact form, you'll get:

$x + \frac{1}{x} + 1 = \frac{3 + \sqrt{5}}{2} + \frac{2}{3 + \sqrt{5}} + 1$

$= \frac{\left(3 + \sqrt{5}\right) \left(3 + \sqrt{5}\right) + 2 \cdot 2 + 2 \left(3 + \sqrt{5}\right)}{2 \left(3 + \sqrt{5}\right)}$

$= \frac{9 + 6 \sqrt{5} + 5 + 4 + 6 + 2 \sqrt{5}}{6 + 2 \sqrt{5}} = \frac{24 + 8 \sqrt{5}}{6 + 2 \sqrt{5}}$

$= \frac{4 \left(6 + 2 \sqrt{5}\right)}{6 + 2 \sqrt{5}} = \frac{4 \cancel{\left(6 + 2 \sqrt{5}\right)}}{\cancel{6 + 2 \sqrt{5}}} = 4$

Jul 23, 2015

Multiply all terms by $x$; rearrange and solve as a standard quadratic to get:
$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{3 \pm \sqrt{5}}{2}$

#### Explanation:

$x + \frac{1}{x} + 1 = 4$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} + 1 + 1 x = 4 x$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$${x}^{2} - 3 x + 1 = 0$

$\Rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$x = (3+-sqrt((-3)^2-4(1)(1)))/(2(1)
$\textcolor{w h i t e}{\text{XXXX}}$$\textcolor{w h i t e}{\text{XXXX}}$$x = \frac{3 \pm \sqrt{5}}{2}$