# Clarification on forming a quadratic equation given the roots

I am confused about the roots and how they can be used to construct the original quadratic equation.

If I am given the roots of the quadratic equations as $$2$$ and $$3$$ I can generate the original equation as $$(x-2)(x-3)=0$$. Is this right?

Now Let's say the equation is $$ax^2+bx+c=0$$ and the roots for this equation are P and Q..

So shouldn't the quadratic equation be obtained as $$(x-P)(x-Q)=0$$, instead in the texts it's mentioned as "$$a(x-P)(x-Q)=0$$"

Isnt the coefficient of $$x^2$$ taken care of while finding the roots?why should we multiply it again?

The text is right in that the $$x^2$$ coefficient is $$a$$ so one needs the factor $$a$$ in order to rewrite the original quadratic. But if you just want some quadratic with given roots your version (without the factor $$a$$) works also.