I just watched a video that teaches a trick to solve some quadratic equations faster:
Suppose we have $3x^2-152x+100=0$ It takes a lot of time to solve it
by finding discriminant because we have to calculate
$152^2$ and so on. we divide $3x^2$ by $3$ and multiply $100$ by $3$
and we get: $x^2-152x+300=0$ we can solve it easily by factoring
$(x-150)(x-2)=0$ then we divide the roots by $3$ so the roots of
original quadratic are $\frac{150}3$ and $\frac23$
It is the first time I see this trick. so is it a known method?
And How we can prove this method works mathematically?
Best Answer
Nice trick, I never saw it.
$$ax^2+bx+c=0\iff a^2x^2+abx+ac=0\iff (ax)^2+b(ax)+ac=0.$$
So you solve for $ax$ and divide by $a$.
You can combine with another trick that I like, when $b$ is even or a factor $2$ can be pulled out (this is a frequent situation):
$$x^2+2bx+c=0\iff x=-b\pm\sqrt{b^2-c}.$$