// maths › Other Useful Identities
Expand and verify (x−a)(x−b) = x² − (a+b)x + ab.
(x-a)(x-b) = x² - (a+b)x + ab
A mind behind this: François Viète 1540–1603
Whenever a quadratic has two positive roots, it factors as (x−a)(x−b) where a and b are those roots. This is the everyday case in solving equations: if x²−5x+6 = 0 factors as (x−2)(x−3) = 0, the solutions are x=2 and x=3. The minus signs mean the roots are the positive numbers a and b themselves, so this form directly reveals where a parabola crosses the x-axis on the positive side.
Expanding gives x² − ax − bx + ab = x² − (a+b)x + ab. Both cross-terms are negative (each x times a negative), so the middle coefficient is −(a+b). But the constant is (−a)(−b) = +ab, two negatives making a positive. So a quadratic with a negative middle term and positive constant has two positive roots - a quick diagnostic when you are factoring.
Look at the constant and middle term. Positive constant means the two roots have the same sign; negative middle term then means both are positive, so you use (x−a)(x−b). If the constant were negative, the roots would have opposite signs. This sign analysis lets you predict the factored form before you find the actual numbers, which speeds up factoring enormously.
Anywhere a quadratic model has two positive solutions: the two times a projectile is at a given height, the two price points that yield a target profit, the two dimensions giving a required area. Factoring as (x−a)(x−b) hands you both solutions directly. It is the same identity as (x+a)(x+b) with sign-flipped roots, and together they cover quadratics with positive roots.
It can be, when two numbers are written as offsets below a base. For example 47×48 with x=50: (50−3)(50−2) = 2500 − 5·50 + 6 = 2500 − 250 + 6 = 2256. The identity organises the arithmetic into an easy square, a quick middle term, and a small product. But its main value remains factorising and solving quadratics.