Essential Mathematics — 1.0 Numbers and Algebra

1.3 Quadratic Equations I (Target age: 15)

Sub-sub-strands (identify & outline)

Formation of quadratic expressions — create algebraic (quadratic) expressions from given situations (areas, products, patterns).
Factorisation and solving quadratic equations — factorise quadratic expressions and use factorisation to solve quadratic equations.
Applications of quadratic equations — model and solve simple real-life problems using quadratic expressions/equations (area, revenue, projectile motion).

Specific learning outcomes (students should be able to):

(b) form algebraic expressions from different situations;
(c) form quadratic expressions and equations from real-life situations;
(d) factorise quadratic expressions in different situations;
(e) solve quadratic equations by factorisation;
(f) recognise the use of quadratic expressions in real-life situations.

1. Formation of quadratic expressions

Quadratic expressions are polynomials where the highest power of the variable is 2, e.g. x² + 5x + 6. They often appear when you multiply two linear expressions or when calculating area.

Example (rectangular garden):
A farmer's rectangular garden has length (x + 3) m and width (x − 1) m. The area A is:

A = (x + 3)(x − 1) = x² + 2x − 3

If the area is 60 m², form and solve the equation: (x + 3)(x − 1) = 60 → x² + 2x − 63 = 0.

2. Factorisation methods

Common methods:

Take out the greatest common factor (GCF).
Factor simple trinomials: x² + bx + c = (x + p)(x + q) where p + q = b and pq = c.
Factor when coefficient of x² is not 1: ax² + bx + c = (mx + p)(nx + q).
Special forms: difference of squares (a² − b² = (a − b)(a + b)), perfect square trinomials (a² ± 2ab + b²).

Worked examples
1) x² + 5x + 6 = (x + 2)(x + 3)
2) 2x² + 7x + 3 = (2x + 1)(x + 3) (check: 2x·x = 2x², outer+inner = 6x + x = 7x)
3) x² − 9 = (x − 3)(x + 3) (difference of squares)
4) x² + 6x + 9 = (x + 3)² (perfect square)

3. Solving quadratic equations by factorisation

To solve: bring equation to one side (standard form = 0), factorise, then use the zero-product property: if (A)(B) = 0 then A = 0 or B = 0.

Example: Solve x² − x − 6 = 0.
Factor: x² − x − 6 = (x − 3)(x + 2) = 0 → x − 3 = 0 or x + 2 = 0 → x = 3 or x = −2.

Example (from the garden): (x + 3)(x − 1) = 60 → x² + 2x − 63 = 0 → (x + 9)(x − 7) = 0 → x = 7 or x = −9. Reject negative; so x = 7 m (length = 10 m, width = 6 m).

4. Applications in real life

Quadratic expressions appear in many contexts useful to students:

Area problems (fields, rooms): product of two linear expressions.
Revenue and profit models: if quantity depends linearly on price then revenue may be quadratic.
Projectile motion (height vs time): h(t) = −(1/2)gt² + vt + h₀ (approximate g ≈ 9.8 m/s²; many exercises use 5 or 10 for simplicity).
Physical patterns and optimisation problems (maximum area, maximum revenue).

Simple revenue example (market):
Suppose customers decrease by 2 for every KSh 1 increase in price. If at KSh 10 there are 50 customers, then number q when price p is: q = 50 − 2(p − 10) = 70 − 2p (an equivalent linear rule). Revenue R = p·q = p(70 − 2p) = −2p² + 70p (a quadratic). The graph is a parabola; the vertex gives the price for maximum revenue.

Visual: a simple parabola (y = x²)

Note: the SVG is a simple sketch to show the U-shape of a parabola. In class, students draw tables of values and sketch by hand or use a graphing calculator.

Suggested learning experiences (classroom activities)

Starter: give several situations (two expressions to multiply, an area problem, a pattern rule). Ask learners to form algebraic expressions; compare answers in pairs.
Group activity: each group receives a word problem (garden, market revenue, simple projectile). Form the quadratic expression and present factorisation and solution method to class.
Hands-on: measure a small rectangular plot (or classroom board area) and write dimensions as expressions; expand and factor the area algebraically.
Practice: worksheets with factorisation types (GCF, trinomials, difference of squares). Include both factorable and non-factorable (leading to use of quadratic formula later).
Use a graphing calculator or free mobile app to plot quadratic functions and identify zeros and vertex (links to real Kenyan examples: optimise small-scale farm area vs fencing cost).
Assessment task: create a short word problem set (3 problems) that lead to quadratic equations; swap with another pair and solve by factorisation.

Practice questions

Factorise: x² + 5x + 6.
Factorise: 3x² + 8x + 4.
A rectangle has sides (x + 4) and (x − 2). If its area is 120 m², find x.
Write the revenue R when price p produces quantity q = 80 − 3p, then state whether R is a quadratic and why.
Solve by factorisation: x² − 4x − 21 = 0.

Answers

(x + 2)(x + 3)
3x² + 8x + 4 = (3x + 2)(x + 2)
(x + 4)(x − 2) = 120 → x² + 2x − 8 − 120 → x² + 2x − 128 = 0 → factors (x + 16)(x − 8) → x = 8 (reject negative if dimension).
R = p(80 − 3p) = −3p² + 80p, yes it is quadratic because p² term appears.
x² − 4x − 21 = (x − 7)(x + 3) → x = 7 or x = −3.

Assessment suggestions and teacher notes

Assess through written problems, group presentations and practical measurements (forming expressions from measured quantities).
Emphasise checking units and rejecting unrealistic roots (negative lengths).
Link problems to local contexts (plots of land, market stalls, school projects) to increase relevance for Kenyan students.
Some quadratics are not factorable with integers — prepare students to meet the quadratic formula in later lessons.

Prepared for classroom use — fits the needs of Kenya secondary learners (age ~15). Teachers may adapt numbers and contexts to local examples (market prices, farm sizes, sports projectiles).

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Topics

Grade 10 essential mathematics Numbers and Algebra – Quadratic Equations I Notes