What does the quadratic formula really do?

Question

I am trying to factorize an expression in terms of x: $3x^2 + 8x + 2$. But I know that in order to do that I have to make it equal to zero. If I find the roots of that equation then equal to zero using the quadratic formula I obtain $2$ surds : $\frac{-4+\sqrt{10}}{3}$ and $\frac{-4-\sqrt{10}}{3}$. So the factorized form will be (x-first root)(x-second root). However, if I multiply that out again I get $x^2 + \frac{8}{3}x + \frac{2}{3}$. So clearly the equation $f(x) = 0$ has been divided by $3$. Is there any way to get the original expression? I know that you could complete the squares but then you wouldn't get it in factorized form.

Answer 1

The factorised form of $ax^2 + bx + c$ will be $a(x-r_1)(x-r_2)$ where $r_1,r_2$ are the two roots (which may be equal). You're forgetting the lead coefficient.

Answer 2

$(x-a)(x-b)$ gives you the equation but with the leading coefficient one. You can use the formula $k(x-a)(x-b)$ where k is that number that when multiplied to the full expression removes the denominator.

Answer 3

The quadratic formula is used to get the roots of a quadratic equation. However, the roots alone doesn't give you the particular equation. This is why you need $3$ conditions to determine the equation, because quadratic equations having different scaling factors of $y$ have the same roots.

---

It can be proved that for a polynomial equation of degree $n$, you need $n + 1$ conditions, and a quadratic function is one of them.