# How do you solve -16x ^ { 2} = 12x ^ { 2} + 24x + 5?

Jan 15, 2018

$x = - \frac{5}{14} , - \frac{1}{2}$

#### Explanation:

In order to solve this equation, first we will have to convert it into the general form, $a {x}^{2} + b x + c = 0$, where $a \ne 0$.

So, $- 16 {x}^{2} = 12 {x}^{2} + 24 x + 5$

$\Rightarrow 28 {x}^{2} + 24 x + 5 = 0$ [Transposing $- 16 {x}^{2}$ to the R.H.S]

Now, we will use the Quadratic Formula or Sridhar Acharya's Formula, to solve for the two roots.

Here, Discriminant = $D$ = ${b}^{2} - 4 a c = {\left(24\right)}^{2} - 4 \cdot 28 \cdot 5$

$= 576 - 560 = 16 > 0$

So, the equation will have two real and distinct roots.

Now, Applying the Formula,

$\alpha$ = $\frac{- b + \sqrt{D}}{2 a} = \frac{- 24 + \sqrt{16}}{2 \cdot 28} = - \frac{20}{56} = - \frac{5}{14}$

and, $\beta$ = $\frac{- b - \sqrt{D}}{2 a} = \frac{- 24 - 4}{2 \cdot 28} = - \frac{28}{56} = - \frac{1}{2}$

So, the two roots of the equation are $- \frac{5}{14}$ and $- \frac{1}{2}$.

Jan 15, 2018

See a solution process below:

#### Explanation:

First, add $\textcolor{red}{16 {x}^{2}}$ to each side of the equation to put the equation in standard form:

$- 16 {x}^{2} + \textcolor{red}{16 {x}^{2}} = 12 {x}^{2} + \textcolor{red}{16 {x}^{2}} + 24 x + 5$

$0 = \left(12 + \textcolor{red}{16}\right) {x}^{2} + 24 x + 5$

$0 = 28 {x}^{2} + 24 x + 5$

$28 {x}^{2} + 24 x + 5 = 0$

Next. we can factor the left side of the equation as:

$\left(14 x + 5\right) \left(2 x + 1\right) = 0$

Now, we can solve each term on the left for $0$:

Solution 1:

$14 x + 5 = 0$

$14 x + 5 - \textcolor{red}{5} = 0 - \textcolor{red}{5}$

$14 x + 0 = - 5$

$14 x = - 5$

$\frac{14 x}{\textcolor{red}{14}} = - \frac{5}{\textcolor{red}{14}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{14}}} x}{\cancel{\textcolor{red}{14}}} = - \frac{5}{14}$

$x = - \frac{5}{14}$

Solution 2:

$2 x + 1 = 0$

$2 x + 1 - \textcolor{red}{1} = 0 - \textcolor{red}{1}$

$2 x + 0 = - 1$

$2 x = - 1$

$\frac{2 x}{\textcolor{red}{2}} = - \frac{1}{\textcolor{red}{2}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} x}{\cancel{\textcolor{red}{2}}} = - \frac{1}{2}$

$x = - \frac{1}{2}$

The Solution Is:

$x = \left\{- \frac{1}{2} , - \frac{5}{14}\right\}$