15a 2 − 3a = 3 − 7a

Good explanation Melody.

I sort of cheat and use the quadratic formula first!  If it returns nice integers or small fractions I can see that it probably factorises nicely!

What do you mean??

15a² - 3a = 3 - 7a

Add 7a to both sides

15a² + 4a = 3

Subtract 3 from both sides

15a² + 4a - 3 = 0

This factorizes nicely:

(3a - 1)*(5a + 3) = 0

So a = 1/3  and a = -3/5

Alan, that is a really good explanation.  But most students would not know how to factorise this quadratic.

I have a method that I will go through but I am wondering how you do it?

$$\\15a^2+4a-3=0\\\\
$You need to find two numbers that multiply to 15*-3=-45$\\
$and add to +4$\\
$Since they multiply to a negative one is $ + $ and the other is $ - \\
$I can see straight off that they are +9 and -5$\\
$Now you split 4a up into -5a+9a $\\\\
15a^2+-5a+9a-3=0\\
15a^2-5a \quad +9a-3=0\\
$Now factorise the pairs$\\
5a(3a-1)+3(3a-1)=0\\
$this factorises to$\\
(5a+3)(3a-1)=0$$

okay, thanks Alan.