Word Problems with Quadratics

≈ 35 min

Quadratic word problems follow the same translate-solve-answer routine as linear ones, with one extra duty: judging the two roots. Mathematics offers both; reality usually accepts one.

Worked example. A rectangular braai area has area $24\ \text{m}^2$ . Its length is 2 m more than its width. Find the width.

Let $w$ = width; length $= w + 2$ :

w(w + 2) = 24 \;\Rightarrow\; w^2 + 2w - 24 = 0 \;\Rightarrow\; (w + 6)(w - 4) = 0

w = -6 \quad \text{or} \quad w = 4

A braai area cannot be $-6$ m wide. Width = 4 m (length 6 m; check $4 \times 6 = 24$ ✓).

Always state why you reject a root — "length must be positive" earns the reasoning mark.

A rectangular braai area has area $24\ \text{m}^2$ and its length is 2 m more than its width. Find the width in metres.

A ball is kicked so its height is $h = 20t - 5t^2$ metres after $t$ seconds. After how many seconds does it land ( $h = 0$ , $t > 0$ )?

Solving a "find the width" problem gives $w = 5$ or $w = -3$ . The correct response is…

Give both: $w = 5$ or $w = -3$

Give $w = 5$ , stating $-3$ is rejected because a width must be positive

Give $w = -3$ since it is smaller

Average them: $w = 1$

Two consecutive whole numbers have a product of 56. What is the smaller number?