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.