# Solve this quadratic equation. Return the answer in 2 decimals ?

Feb 16, 2018

$x = 3.64 , - 0.14$

#### Explanation:

We have $2 x - \frac{1}{x} = 7$

Multiplying both sides by $x$, we get:

$x \left(2 x - \frac{1}{x}\right) = 7 x$

$2 {x}^{2} - 1 = 7 x$

$2 {x}^{2} - 7 x - 1 = 0$

Now we have a quadratic equation. For any $a {x}^{2} + b x + c = 0$, where $a \ne 0 ,$ $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$.

Here, $a = 2 , b = - 7 , c = - 1$

We can input:

$\frac{- \left(- 7\right) \pm \sqrt{{\left(- 7\right)}^{2} - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

$\frac{7 \pm \sqrt{49 + 8}}{4}$

$\frac{7 \pm \sqrt{57}}{4}$

$x = \frac{7 + \sqrt{57}}{4} , \frac{7 - \sqrt{57}}{4}$

$x = 3.64 , - 0.14$

Feb 16, 2018

$x = 3.64 \mathmr{and} x = - 0.14$

#### Explanation:

This is clearly not a comfortable form to work with.
Multiply through by $x$ and re-arrange the equation into the form:

$a {x}^{2} + b x + c = 0$

$2 x \textcolor{b l u e}{\times x} - \frac{1}{x} \textcolor{b l u e}{\times x} = 7 \textcolor{b l u e}{\times x}$

$2 {x}^{2} - 1 = 7 x$

$2 {x}^{2} - 7 x - 1 = 0 \text{ } \leftarrow$ it does not factorise

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

$x = \frac{- \left(- 7\right) \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(2\right) \left(- 1\right)}}{2 \left(2\right)}$

$x = \frac{7 \pm \sqrt{49 + 8}}{4}$

$x = \frac{7 + \sqrt{57}}{4} = 3.64$

$x = \frac{7 - \sqrt{57}}{4} = - 0.14$

Feb 16, 2018

See below...

#### Explanation:

First we need the standard format of $a {x}^{2} + b x + c = 0$

First we multiply all by $x$ to remove the fraction.

$2 x - \frac{1}{x} = 7 \implies 2 {x}^{2} - 1 = 7 x$

Now we move the $7 x$ over by subtracting both sides by $7 x$

$2 {x}^{2} - 1 = 7 x \implies 2 {x}^{2} - 7 x - 1 = 0$

As we want the answers to $2 d . p$ it strongly hints that we need to use the quadratic formula.

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

Now from our equation we know that ...

$a = 2$, $b = - 7$ and $c = - 1$

Now we plug these into our formula, but as we have a $+$ and a $-$ we have to do it twice.

$x = - \left(- 7\right) + \frac{\sqrt{{\left(- 7\right)}^{2} - 4 \left(2\right) \left(- 1\right)}}{2 \left(2\right)}$
$x = - \left(- 7\right) - \frac{\sqrt{{\left(- 7\right)}^{2} - 4 \left(2\right) \left(- 1\right)}}{2 \left(2\right)}$

Now we put each one into our calculator and round to $2 d . p .$

$\therefore x = - 0.14 , x = 3.64$

Both to $2 d . p$