# What are the approximate solutions of 2x^2 + x = 14 rounded to the nearest hundredth?

Aug 28, 2017

color(green)(x=2.41 or $\textcolor{g r e e n}{x = - 2.91} \textcolor{w h i t e}{\text{xxx}}$(both to the nearest hundrdeth.

#### Explanation:

Re-writing the given equation as
$\textcolor{w h i t e}{\text{XXX}} \textcolor{red}{2} {x}^{2} + \textcolor{b l u e}{1} x \textcolor{g r e e n}{- 14} = 0$
and applying the quadratic formula:
color(white)("XXX")x=(-color(blue)1+-sqrt(color(blue)1^2-4 * color(red)2 * color(green)(""(-14))))/(2 * color(red)2)

$\textcolor{w h i t e}{\text{XXXx}} = \frac{- 1 \pm \sqrt{113}}{4}$

with the use of a calculator (or, im my case I used a spreadsheet)
$\textcolor{w h i t e}{\text{XXX")x ~~ 2.407536453color(white)("xxx")orcolor(white)(} \times x '} x \approx - 2.9075366453$

Rounding to the nearest hundredths gives the results in the "Answer" (above)

Aug 28, 2017

See a solution process below:

#### Explanation:

First, we can subtract $\textcolor{red}{14}$ from each side of the equation to put the equation in standard form while keeping the equation balanced:

$2 {x}^{2} + x - \textcolor{red}{14} = 14 - \textcolor{red}{14}$

$2 {x}^{2} + x - 14 = 0$

We can now use the quadratic equation to solve this problem.

The quadratic formula states:

For $\textcolor{red}{a} {x}^{2} + \textcolor{b l u e}{b} x + \textcolor{g r e e n}{c} = 0$, the values of $x$ which are the solutions to the equation are given by:

$x = \frac{- \textcolor{b l u e}{b} \pm \sqrt{{\textcolor{b l u e}{b}}^{2} - \left(4 \textcolor{red}{a} \textcolor{g r e e n}{c}\right)}}{2 \cdot \textcolor{red}{a}}$

Substituting:

$\textcolor{red}{2}$ for $\textcolor{red}{a}$

$\textcolor{b l u e}{1}$ for $\textcolor{b l u e}{b}$

$\textcolor{g r e e n}{- 14}$ for $\textcolor{g r e e n}{c}$ gives:

$x = \frac{- \textcolor{b l u e}{1} \pm \sqrt{{\textcolor{b l u e}{1}}^{2} - \left(4 \cdot \textcolor{red}{2} \cdot \textcolor{g r e e n}{- 14}\right)}}{2 \cdot \textcolor{red}{2}}$

$x = \frac{- \textcolor{b l u e}{1} \pm \sqrt{1 - \left(- 112\right)}}{4}$

$x = \frac{- \textcolor{b l u e}{1} \pm \sqrt{1 + 112}}{4}$

$x = \frac{- \textcolor{b l u e}{1} - \sqrt{1 + 112}}{4}$ and $x = \frac{- \textcolor{b l u e}{1} + \sqrt{1 + 112}}{4}$

$x = \frac{- \textcolor{b l u e}{1} - \sqrt{113}}{4}$ and $x = \frac{- \textcolor{b l u e}{1} + \sqrt{113}}{4}$

$x = \frac{- \textcolor{b l u e}{1} - 10.6301}{4}$ and $x = \frac{- \textcolor{b l u e}{1} + 10.6301}{4}$

$x = - \frac{11.6301}{4}$ and $x = \frac{9.6301}{4}$

$x = - 2.91$ and $x = 2.41$ rounded to the nearest hundredth.