The picture shows how 5 friends shake hands: every pair shakes hands exactly once and each line is one handshake. At Meera's birthday party, every pair of guests also shook hands once, and there were 45 handshakes in all. How many guests were there?

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).