Roots and Coefficients: Sum, Product, and Building an Equation from Given Roots

Once you have the root formula, adding and multiplying the two roots $\alpha = \dfrac{-b+\sqrt{\Delta}}{2a}$ and $\beta = \dfrac{-b-\sqrt{\Delta}}{2a}$ produces two remarkably clean identities — the square-root parts cancel in the sum and simplify in the product:

In words: the sum of the roots is minus the coefficient of $x$ divided by the coefficient of $x^2$, and the product of the roots is the constant term divided by the coefficient of $x^2$. When $a=1$ this is even simpler — the sum is $-b$ and the product is just $c$.

These two numbers are enough to rebuild the whole equation, because $ax^2+bx+c = a(x-\alpha)(x-\beta) = a\big[x^2-(\alpha+\beta)x+\alpha\beta\big]$. So any quadratic equation with roots $\alpha,\beta$ can be written as

This is exactly the tool the syllabus calls "forming a quadratic equation given its roots" — you don't need to know $a,b,c$ individually at all, only the sum and the product of the two numbers you want as roots. It also runs in reverse: given a quadratic equation, you instantly know the sum and product of its roots without ever solving it, which is often all a problem actually needs (e.g. finding $\alpha^2+\beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$, or $\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$, purely from $a,b,c$).

A related, useful fact: two quadratic equations $a_1x^2+b_1x+c_1=0$ and $a_2x^2+b_2x+c_2=0$ share a common root exactly when $(c_1a_2-c_2a_1)^2 = (a_1b_2-a_2b_1)(b_1c_2-b_2c_1)$ — a condition you can check without ever solving either equation.

Worked example. Form the quadratic equation whose roots are $3+\sqrt5$ and $3-\sqrt5$.

Let $\alpha = 3+\sqrt5,\ \beta = 3-\sqrt5$. Then

$\alpha+\beta = 6, \qquad \alpha\beta = 3^2 - (\sqrt5)^2 = 9-5=4.$ …