Sum and product of the roots

A pattern in the coefficients

The equation \(x^2 - 9x + 20 = 0\) has roots \(4\) and \(5\). Notice:

• their sum is \(9\), the opposite of the \(x\) coefficient;
• their product is \(20\), the constant term.

This isn't luck. Add the two roots from the quadratic formula, and the square roots cancel; multiply them, and the difference of squares removes the square root. Either way, a simple formula is left.

Here's the sum, written out:

For the product, the numerators multiply to \(b^2 - (b^2 - 4ac) = 4ac\), so the product is \(\frac{4ac}{4a^2} = \frac{c}{a}\).

Factoring any quadratic

This works even when the "guess two numbers" method is hard, such as when \(a \neq 1\) or the roots are fractions.

Writing an equation from its roots

Divide \(ax^2 + bx + c = 0\) by \(a\) and you get \(x^2 + \frac{b}{a}x + \frac{c}{a} = 0\). In terms of the sum \(S\) and product \(P\) of the roots:

Finding a missing coefficient

Practice