sum and product of roots

The sum and product of roots are two slick shortcuts that connect a quadratic's two roots directly to its coefficients — without solving for the roots first. They let you check answers fast, build an equation from desired roots, or find one root when you already know the other.

For ax^2 + bx + c = 0 with roots r and s: the sum r + s = -b/a, and the product r·s = c/a. You can see why by writing the quadratic as a(x - r)(x - s) and expanding: matching the x-term gives the sum, matching the constant gives the product.

A neat consequence: any quadratic with leading coefficient 1 can be written as x^2 - (sum)x + (product). So if you want a quadratic whose roots are 2 and 5, the sum is 7 and the product is 10, giving x^2 - 7x + 10 = 0 immediately. These relations are the two-root special case of Vieta's formulas.