# How do you Identify the solutions to the equation 2x^2+5x=3 when you solve by factoring?

Aug 20, 2016

$x = - 3 \mathmr{and} x = \frac{1}{2}$

#### Explanation:

Firstly we note that it is a quadratic.( has ${x}^{2}$), so we make it equal to 0.

$2 {x}^{2} + 5 x - 3 = 0 \text{ factorise}$

Find factors of 2 and 3 which subtract to give 5.
The signs will be different, there must be more positives.

As both 2 and 3 are prime numbers there are only 2 possibilities.
Cross multiply the factors:

$\textcolor{w h i t e}{\times \times} \left(2\right) \text{ } \left(3\right)$
color(white)(xx.x) (1" " 3)rArr" " 2xx3 = 6
color(white)(xx.x) (2" " 1)rArr" " 1xx1 = 1 " "6-1 =5

we have the correct factors, now put in the correct signs.

$\textcolor{w h i t e}{\times . x} \left(2\right) \text{ } \left(3\right)$
color(white)(xx.x) (1" " +3)rArr" " 2xx+3 = +6
color(white)(xx.x) (2" "- 1)rArr" " 1xx-1 = -1 " "+6-1 =+5

The factors are:

$\left(x + 3\right) \left(2 x - 1\right) = 0$

One of the factors must be 0 to give a product of 0.

$x + 3 = 0 \Rightarrow x = - 3$
$2 x - 1 = 0 \Rightarrow x = \frac{1}{2}$