The width $x$ metres of a rectangular vegetable patch satisfies $x^2+5x-126=0$, and the quadratic formula gives $x=9$ and $x=-14$.
Which value is the width, and why?

$D=25+504=529$, so $x=\frac{-5\pm 23}{2}=9$ or −14. Both numbers satisfy the equation algebraically — substituting −14 gives 196 − 70 − 126 = 0 — but a length in metres must be positive, so −14 is rejected on physical grounds and the width is 9 m.