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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?

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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.

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