# How do you solve 15a^2 - 3a = 3 - 7a by factoring?

Aug 21, 2015

The solutions are

color(blue)(a=1/3

 color(blue)(a=-3/5

#### Explanation:

15a^2−3a=3−7a

15a^2−3a+7a-3=0

$15 {a}^{2} + 4 a - 3 = 0$

We can Split the Middle Term of this expression to factorise it and thereby find solutions.

In this technique, we need to think of 2 numbers such that:
${N}_{1} \cdot {N}_{2} = a \cdot c = 15 \cdot - 3 = - 45$
AND
${N}_{1} + {N}_{2} = b = 4$

After trying out a few numbers we get ${N}_{1} = 9$ and ${N}_{2} = - 5$
$9 \cdot - 5 = - 45$, and $9 + \left(- 5\right) = 4$

$15 {a}^{2} + 4 a - 3 = 15 {a}^{2} + 9 a - 5 a - 3$

$15 {a}^{2} + 9 a - 5 a - 3 = 0$

$3 a \left(5 a + 3\right) - 1 \left(5 a + 3\right) = 0$
$\left(3 a - 1\right) \left(5 a + 3\right) = 0$ is the factorised form of the equation.

Now we equate factors to zero.
3a-1 = 0, color(blue)(a=1/3

5a+3 = 0, color(blue)(a=-3/5