You can already solve a quadratic equation by factorising, completing the square or using the formula. This chapter turns the question around: without solving \(ax^2+bx+c=0\), what do its roots \(\alpha\) and \(\beta\) tell us about the coefficients \(a\), \(b\) and \(c\)?**Sum and product of roots of \(ax^2+bx+c=0\), \(a\ne 0\)**Both results come from comparing coefficients: dividing by \(a\) and factorising gives \(x^2+\dfrac{b}{a}x+\dfrac{c}{a}=(x-\alpha)(x-\beta)=x^2-(\alpha+\beta)x+\alpha\beta\).**Building a quadratic from its roots**Reversing the comparison, a quadratic equation with roots \(\alpha\) and \(\beta\) is \(x^2-(\alpha+\beta)x+\alpha\beta=0\), that is, \(x^2-(\text{sum of roots})\,x+(\text{product of roots})=0\).Rearrange into the form \(ax^2+bx+c=0\) before reading off \(a\), \(b\) and \(c\) — signs included. The sum of roots carries a negative sign: \(\alpha+\beta=-\frac{b}{a}\).

You can already solve a quadratic equation by factorising, completing the square or using the formula. This chapter turns the question around: without solving \(ax^2+bx+c=0\), what do its roots \(\alpha\) and \(\beta\) tell us about the coefficients \(a\), \(b\) and \(c\)?

From solving to structure

**Sum and product of roots of \(ax^2+bx+c=0\), \(a\ne 0\)**

Both results come from comparing coefficients: dividing by \(a\) and factorising gives \(x^2+\dfrac{b}{a}x+\dfrac{c}{a}=(x-\alpha)(x-\beta)=x^2-(\alpha+\beta)x+\alpha\beta\).

Reversing the comparison, a quadratic equation with roots \(\alpha\) and \(\beta\) is \(x^2-(\alpha+\beta)x+\alpha\beta=0\), that is, \(x^2-(\text{sum of roots})\,x+(\text{product of roots})=0\).

Rearrange into the form \(ax^2+bx+c=0\) before reading off \(a\), \(b\) and \(c\) — signs included. The sum of roots carries a negative sign: \(\alpha+\beta=-\frac{b}{a}\).

Result	Formula	Example: \(2x^2-8x+6=0\)
Sum of roots	\(\alpha+\beta=-\dfrac{b}{a}\)	\(\alpha+\beta=-\dfrac{-8}{2}=4\)
Product of roots	\(\alpha\beta=\dfrac{c}{a}\)	\(\alpha\beta=\dfrac{6}{2}=3\)

Sum and Product of Roots of a Quadratic Equation

Sum and Product of Roots of a Quadratic Equation