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Learning Resources

Quadratic Expressions and Equations

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 10 Pathway: N/A

First Principles

Learning Objective: Understand the structure of quadratic expressions, expand binomial products geometrically, and solve quadratic equations by factorisation and formula.

Core Concept: A quadratic expression represents quantities in two dimensions (like area). Written in standard form \(ax^2 + bx + c\) (where \(a \neq 0\)), it arises whenever two linear quantities multiply, such as the length and width of a maize shamba.

1. Concrete Scenario: The Shamba Extension

Imagine Farmer Otieno has a square farm plot of side length \(x\) metres. He decides to expand it by adding \(a\) metres along the length and \(b\) metres along the width. The new area is \((x + a)(x + b)\). When subdivided, the expanded shamba consists of four distinct sections:

  • A square plot of \(x \times x = x^2\)
  • Two rectangular extension strips of areas \(ax\) and \(bx\)
  • A corner plot of area \(a \times b\)

Combining these gives the total area: \[\text{Area} = x^2 + ax + bx + ab = x^2 + (a+b)x + ab\]

2. Factorisation as Reverse Expansion

To factorise \(x^2 + bx + c\), we reverse this process: we search for two integers \(p\) and \(q\) whose product is \(c\) (the corner area) and whose sum is \(b\) (the combined strip width). Once found, we write the expression as \((x + p)(x + q)\).

Interactive Visualizer: Shamba Area Model

Adjust the sliders for \(a\) and \(b\) to see how the geometric area partitions and rebuilds the algebraic terms:

x²
+3x
+2x
+6
Factored Form: (x + 2)(x + 3)
Expanded Form: x² + 5x + 6

Key Formulas

Standard Form: \[ax^2 + bx + c = 0 \quad (a \neq 0)\] All quadratic equations can be rearranged into this standard form.
Factorisation Identity: \[(x + p)(x + q) = x^2 + (p + q)x + pq\] Where the middle coefficient is the sum \(b = p + q\) and the constant is the product \(c = pq\).
Difference of Two Squares (DOTS): \[a^2 - b^2 = (a - b)(a + b)\] Perfect Square Expansions: \[(a \pm b)^2 = a^2 \pm 2ab + b^2\]
The Quadratic Formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
The Discriminant (\(\Delta\)): \[\Delta = b^2 - 4ac\]
  • If \(\Delta > 0\): Two distinct real roots
  • If \(\Delta = 0\): One repeated real root (equal roots)
  • If \(\Delta < 0\): No real roots (two complex conjugate roots)

Worked Examples

Example 1: Solving by Factorisation (Monotonous/Easy)

Problem: Solve \(x^2 - 7x + 12 = 0\).

  1. Find two numbers: We need two numbers that multiply to \(+12\) and add to \(-7\). Listing factors of 12 gives \(-3\) and \(-4\), since \((-3)(-4) = 12\) and \((-3) + (-4) = -7\).
  2. Factorise: \((x - 3)(x - 4) = 0\).
  3. Apply Zero-Product Property: Either \(x - 3 = 0\) or \(x - 4 = 0\).
  4. Solve: \(x = 3\) or \(x = 4\).
Solution: \(x = 3\) or \(x = 4\)
Example 2: Non-Monic Quadratic by Grouping / Formula (Medium)

Problem: Solve \(2x^2 + 5x - 3 = 0\).

  1. Identify coefficients: \(a = 2, b = 5, c = -3\).
  2. Find product and sum: Product \(ac = 2 \times (-3) = -6\), Sum \(b = 5\). The factors are \(6\) and \(-1\).
  3. Split the middle term: \[2x^2 + 6x - x - 3 = 0\]
  4. Factor by grouping: \[2x(x + 3) - 1(x + 3) = 0 \implies (2x - 1)(x + 3) = 0\]
  5. Solve each linear factor: \[2x - 1 = 0 \implies x = \frac{1}{2}\] \[x + 3 = 0 \implies x = -3\]
Solution: \(x = 0.5\) or \(x = -3\)
Example 3: Real-World Shamba Geometry (Hard/Multi-step)

Problem: A rectangular vegetable garden has a length that is \(3\text{ m}\) longer than its width. If the total area of the garden is \(40\text{ m}^2\), find the perimeter of the garden.

  1. Define variables: Let width \(= w\). Then length \(= w + 3\).
  2. Set up equation: \[\text{Area} = w(w + 3) = 40 \implies w^2 + 3w - 40 = 0\]
  3. Factorise: Find two numbers multiplying to \(-40\) and summing to \(3\): \(+8\) and \(-5\). \[(w + 8)(w - 5) = 0\]
  4. Discard non-physical solutions: \(w = -8\) or \(w = 5\). Since length/width cannot be negative, \(w = 5\text{ m}\).
  5. Calculate dimensions & perimeter: \[\text{Width} = 5\text{ m}, \quad \text{Length} = 5 + 3 = 8\text{ m}\] \[\text{Perimeter} = 2(\text{Length} + \text{Width}) = 2(8 + 5) = 26\text{ m}\]
Solution: Perimeter = \(26\text{ m}\)

Common Mistakes

Mistake 1: "Distributing" the square over addition
Incorrect: \((x + 4)^2 = x^2 + 16\)
Correct: \((x + 4)^2 = x^2 + 8x + 16\)
Why it feels right: Students intuitively remember index laws like \((ab)^2 = a^2 b^2\), but addition creates cross-terms \(2ab\) corresponding to the two rectangular strips in the area model.
Mistake 2: Cancelling terms before setting equation to zero
Incorrect: Solving \(x^2 = 5x\) by dividing both sides by \(x\) to get \(x = 5\).
Correct: Rearrange to \(x^2 - 5x = 0 \implies x(x - 5) = 0\), yielding two solutions: \(x = 0\) and \(x = 5\).
Why it feels right: Dividing simplifies the equation quickly, but dividing by \(x\) eliminates the valid root \(x = 0\) (and is illegal if \(x=0\)).
Mistake 3: Fraction denominator error in the quadratic formula
Incorrect: Writing \(x = -b \pm \frac{\sqrt{b^2 - 4ac}}{2a}\)
Correct: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Why it feels right: Students often punch \(-b\) separately into a calculator without grouping the entire numerator inside brackets.

Real World

1. Agriculture & Shamba Optimization

In Nakuru and Eldoret, farmers often fence rectangular sections of land for poultry or horticulture with fixed fencing perimeter. If a farmer has \(60\text{ m}\) of wire mesh to enclose three sides against an existing wall, the area equation is \(A = w(60 - 2w) = -2w^2 + 60w\). Finding the maximum area uses the quadratic vertex.

2. Matatu SACCO Fleet Economics

A transport SACCO operating between Nairobi and Thika finds that ticketing revenue is modeled by \(R(x) = -5x^2 + 1200x\), where \(x\) is ticket price in KSh. Quadratics help determine the break-even points and optimal fare pricing to balance passenger volume with total revenue.

3. Ballistics and Sports

When a goalkeeper at Nyayo Stadium kicks a football into the air, its height in metres after \(t\) seconds is modeled by \(h(t) = -4.9t^2 + 19.6t\). Solving \(h(t) = 0\) reveals the exact time the ball stays airborne before landing.

Practice

Nyambura, a community nurse, is preparing a square medical patch whose area is given by the expression \((x + 3)^2\) square centimetres. If \(x = 3\), what is the area of the patch in square centimetres? (Type only the number, e.g., 42)
Review the concepts above.
Susan, a pharmacist in Nairobi, is arranging medicine containers on a display shelf. The area in square centimetres is represented by \((2x + 3)^2\). If \(x = 4\), calculate the total area of the shelf in square centimetres. (Type only the number, e.g., 42)
Review the concepts above.
Osogo, a carpenter in Kisumu, is constructing a rectangular timber fence for a community garden. The area is \(x^2 - x - 12\) square metres and the length is \((x - 4)\) metres. If \(x = 10\), what is the numerical width of the fence in metres? (Type only the number, e.g., 42)
Review the concepts above.
A matatu operator calculates that his daily profit \(P\) (in thousands of shillings) is modeled by \(P = -x^2 + 10x - 21\), where \(x\) is the number of round trips completed. What is the smaller number of trips for which his profit is zero (the break-even point)? (Type only the number, e.g., 42)
Review the concepts above.
Solve the quadratic equation \(2x^2 - 4x - 30 = 0\) for its positive root. (Type only the number, e.g., 5)
Review the concepts above.
A matatu courier service determines the cost \(C\) (in shillings) for an express trip is \(C = 5000 + 1500x + 250x^2\), where \(x\) is the number of extra delivery zones covered. If the total charge for a trip is \(6750\) shillings, how many extra delivery zones \(x\) were covered? (Type only the number, e.g., 2)
Review the concepts above.