From a square sheet of side 30 cm, a square of side $x$ cm is cut from each corner (red, dashed) and the sides are folded up along the dashed lines to make an open box. The base of the box has area $196{\text{cm}}^2$. Find $x$.

Many real-world problems lead to quadratics:

**Number problems**: 'Find two consecutive integers whose product is 132.' Let $n,n+1$; $n(n+1)=132$ → $n^2+n-132=0$ → $n=11$ or −12. Both valid (or pick positive).

**Area problems**: 'A garden's length is 5 m more than its width. Area is $84{\text{m}}^2$. Find dimensions.'
Let width = $x$. Length = $x+5.x(x+5)=84$ → $x^2+5x-84=0$ → $x=7$ (rejecting negative).

**Speed/time/distance**: 'A train takes 1 hour less if its speed is increased by 10 km/h. Original speed?' → typical setup: time1 − time2 = 1 → distance/speed equations → quadratic in speed.

**Always**: reject negative or extraneous roots if context demands positive (lengths, ages, speeds).