{"id":63445,"date":"2026-06-10T22:46:05","date_gmt":"2026-06-10T14:46:05","guid":{"rendered":"https:\/\/blog-admin.thethinkacademy.com\/?p=63445"},"modified":"2026-06-10T22:46:07","modified_gmt":"2026-06-10T14:46:07","slug":"quadratic-word-problems-grade-10-cayley","status":"publish","type":"post","link":"https:\/\/blog-admin.thethinkacademy.com\/blog\/2026\/06\/10\/quadratic-word-problems-grade-10-cayley\/","title":{"rendered":"Quadratic Word Problems: Every Type Explained with Worked Examples (Grade 10 and Cayley Contest)"},"content":{"rendered":"\n

Quadratic word problems are where the algebra becomes real. Students who can expand and factor a quadratic expression often struggle when the same mathematics is wrapped in a context \u2014 a projectile, a revenue model, a rectangle with given area. The problem looks unfamiliar. The equation is not obvious. This guide covers every type of quadratic word problem that appears in Ontario Grade 10 math and the Cayley Contest<\/a>, with a clear method for each and fully worked examples throughout.<\/p>\n\n\n\n

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The four types of quadratic word problems<\/h2>\n\n\n\n

Every quadratic word problem in Grade 10 Ontario math and the Cayley Contest falls into one of four categories. Identifying the type before setting up the equation is the most important step.<\/p>\n\n\n\n

Type<\/th>What you are finding<\/th>Key phrase in the problem<\/th><\/tr><\/thead>
Projectile \/ motion<\/td>Maximum height, time in air, when object hits ground<\/td>‘thrown’, ‘launched’, ‘ball’, ‘height’, ‘falls’<\/td><\/tr>
Area and geometry<\/td>Dimensions of a shape given area or perimeter<\/td>‘rectangle’, ‘area’, ‘width’, ‘length’, ‘border’<\/td><\/tr>
Revenue and optimisation<\/td>Maximum profit, optimal price, number sold<\/td>‘revenue’, ‘profit’, ‘price’, ‘sales’, ‘maximise’<\/td><\/tr>
Number problems<\/td>Two unknowns with a quadratic relationship<\/td>‘product’, ‘sum’, ‘consecutive integers’, ‘differ by’<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Once you have identified the type, the setup process is the same in every case: define a variable, write the quadratic equation, and solve using the appropriate method.<\/p>\n\n\n\n

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A general method for any quadratic word problem<\/h2>\n\n\n\n

Step 1: Read carefully and identify the type.<\/strong> What is the problem actually asking you to find? A time? A dimension? A price? A pair of numbers?<\/p>\n\n\n\n

Step 2: Define your variable.<\/strong> Let x = (whatever the unknown quantity is). Be specific \u2014 ‘let x = the width of the rectangle in centimetres’, not just ‘let x = width’.<\/p>\n\n\n\n

Step 3: Write the equation.<\/strong> Express the relationship described in the problem as a quadratic equation. This is the hardest step and the one most worth practising.<\/p>\n\n\n\n

Step 4: Solve the equation.<\/strong> Use factoring, the quadratic formula, or completing the square depending on what the problem requires and what form the equation takes.<\/p>\n\n\n\n

Step 5: Interpret the solution.<\/strong> A quadratic equation usually produces two solutions. Check both against the context. Negative dimensions, negative prices, and times before t = 0 are typically inadmissible \u2014 eliminate them and state why.<\/p>\n\n\n\n

Step 6: Write a conclusion.<\/strong> Answer the question asked. Include units.<\/p>\n\n\n\n

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Type 1: Projectile and motion problems<\/h2>\n\n\n\n

Projectile problems use the relationship between height and time. In Ontario Grade 10 math<\/a>, the standard model is:<\/p>\n\n\n\n

h = \u22124.9t\u00b2 + v\u2080t + h\u2080<\/strong><\/p>\n\n\n\n

Where:<\/p>\n\n\n\n