Mathematics • Year 9 • Unit 2 • Lesson 18
Quadratic Equations in Context
Four real situations that all turn into a quadratic equation: a square garden, a rectangular pen, a tossed phone, a number puzzle. Set up, factor, solve, then decide which solution is physically meaningful.
1. Word problems
For each: set up the equation, factor or inspect, solve, then check which root is physically meaningful. 3 marks each
1.1 — Square garden. A square garden has area $64$ m². Let the side length be $x$ metres.
(a) Write the quadratic equation that says "area is $64$".
(b) Solve it by inspection.
(c) Which root is physically meaningful, and why is the other one rejected?
1.2 — Rectangular chicken pen. A rectangular chicken pen has length $(x + 3)$ metres and width $(x - 2)$ metres. Its area is $50$ m².
(a) Write an expression for the area as $(x + 3)(x - 2)$ and expand.
(b) Set the area equal to $50$, rearrange to get $x^2 + bx + c = 0$ form, and factor.
(c) Solve, then reject the root that gives a non-positive length or width.
1.3 — Tossed phone. A phone is tossed and its height (m) above the ground after $t$ seconds is $h = -5t^2 + 10t$. We want to know when it hits the ground ($h = 0$).
(a) Set $h = 0$ and factor (common factor first).
(b) State both solutions.
(c) Which solution is the landing time, and what does the other one represent physically?
1.4 — Two consecutive numbers. Two consecutive positive whole numbers multiply to give $72$. Let the smaller one be $x$.
(a) Write the equation $x(x + 1) = 72$ and rearrange to $x^2 + x - 72 = 0$.
(b) Factor and solve.
(c) Which root is the smaller consecutive number, and what are the two numbers? Why is the other root rejected?
2. Explain your thinking
Use full sentences. 4 marks
2.1 A classmate solves $x^2 = 36$ and writes "$x = 6$" as the final answer. In your own words, explain (i) what crucial part of the answer is missing, (ii) why both $6$ and $-6$ are valid solutions of $x^2 = 36$, and (iii) give an example of a real-world problem where only the positive solution is physically meaningful (so the missing $-6$ wouldn't matter).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Square garden
(a) Equation: $x^2 = 64$.
(b) Inspection: $x = \pm \sqrt{64} = \pm 8$.
(c) Physically meaningful: $x = 8$ m (a side length must be positive). $x = -8$ is rejected because length cannot be negative.
1.2 — Chicken pen
(a) Area $= (x + 3)(x - 2) = x^2 + x - 6$.
(b) Set $= 50$: $x^2 + x - 6 = 50 \Rightarrow x^2 + x - 56 = 0$. Factor: product $-56$, sum $1$: try $8$ and $-7$. So $(x + 8)(x - 7) = 0$.
(c) $x = -8$ or $x = 7$. Reject $x = -8$ because length $= x + 3 = -5$ (negative) and width $= x - 2 = -10$ (also negative) — neither makes sense for a real pen. Accept $x = 7$: length $= 10$ m, width $= 5$ m, area $= 50$ m² ✓.
1.3 — Tossed phone
(a) $-5t^2 + 10t = 0 \Rightarrow -5t(t - 2) = 0$.
(b) $t = 0$ or $t = 2$.
(c) Landing time: $t = 2$ s. The other solution, $t = 0$, represents the LAUNCH moment (when the phone was at height $0$ before being thrown up). It's not "rejected" in the impossible sense — it's just not the answer the question wants (it's the START, not the END).
1.4 — Consecutive numbers
(a) $x(x + 1) = 72 \Rightarrow x^2 + x - 72 = 0$.
(b) Product $-72$, sum $1$: try $9$ and $-8$. Factored: $(x + 9)(x - 8) = 0$. Roots: $x = -9$ or $x = 8$.
(c) Smaller consecutive number: $x = 8$. The two numbers are $8$ and $9$ (check: $8 \times 9 = 72$ ✓). Reject $x = -9$ because the question specifies POSITIVE whole numbers. (Note: $-9 \times -8 = 72$ also works mathematically, but the problem said "positive".)
2.1 — Explain your thinking (sample response)
My classmate is missing the second solution: the equation $x^2 = 36$ has TWO real solutions, $x = 6$ and $x = -6$, not just one. Both work because $6^2 = 36$ AND $(-6)^2 = 36$ — squaring removes the sign, so any number and its negative both square to the same positive value. The correct answer is $x = \pm 6$. However, in real-world problems we often reject the negative solution because the variable represents something physical that can't be negative — for example, if $x$ is the side length of a square of area $36$ m², only $x = 6$ m is meaningful (a side length can't be $-6$ m). In a number-puzzle context like "what number squared is $36$?" both answers are valid; in a measurement context the negative is rejected.
Marking: 1 mark for "missing the $-6$"; 1 mark for "both signs squared give the same positive"; 1 mark for a real-world example where only the positive matters; 1 mark for clear, full-sentence writing.