2️⃣ Mastering Equations

1 Diagnostic Activity

Answer the following questions without a calculator. They check the prerequisite knowledge for this module.

1. Factor \(x^2 + 4x\).
2. Expand \(2(3x - 5)\).
3. Solve \(5x - 8 = 12\).

NoteCheck your answers

1. \(x^2 + 4x = x(x + 4)\)
2. \(2(3x - 5) = 6x - 10\)
3. \(5x - 8 = 12 \Rightarrow 5x = 20 \Rightarrow x = 4\)

Use your results to identify topics needing attention.

2 Opening Problem

A small café has fixed monthly costs of €2,000 and variable costs of €3 per unit sold. Each unit sells at €8. How many units must the café sell each month to cover all its costs?

Write down your estimate before continuing.

3 Core Concepts

3.1 Algebraic Transformations

Algebraic transformations are the set of rules and techniques used to change the form of an algebraic expression without changing its value.

These operations let us simplify, expand, factor, or rewrite expressions in order to:

1. make them easier to manipulate;
2. reveal a structure (such as a notable identity or a common factor); or
3. make it easier to solve equations.

3.2 Factoring

Factoring consists in rewriting an expression — for example \(C(x) = x^{2} - 3x\) — in another form that contains a product. On the right-hand side, the variable \(x\) appears in both \(x^{2}\) and \(3x\), so we can use \(x\) as a common factor. The factored form becomes

\[ C(x) = x(x - 3). \]

Factoring makes the behaviour of the function easier to read: we see at once that the function is zero when \(x = 0\) or \(x = 3\).

TipSteps of factoring

1. Identify the greatest common factor.
2. Divide each term by that factor.
3. Rewrite the expression as a product.

Exercises — factor

CautionBasic common factor (click for solutions)

1. \(4x + 8\) → GCF \(4\): \(\;4(x + 2)\)
2. \(-21x - 14\) → GCF \(-7\): \(\;-7(3x + 2)\)
3. \(35x + 49\) → GCF \(7\): \(\;7(5x + 7)\)
4. \(144 + 108x\) → GCF \(36\): \(\;36(4 + 3x)\)
5. \(x^{2} + 4x\) → GCF \(x\): \(\;x(x + 4)\)
6. \(8x^{2} + 4x\) → GCF \(4x\): \(\;4x(2x + 1)\)
7. \(x^{3} + x^{2}\) → GCF \(x^{2}\): \(\;x^{2}(x + 1)\)
8. \(60x^{3} + 90x^{2}\) → GCF \(30x^{2}\): \(\;30x^{2}(2x + 3)\)
9. \(20xy + 30x\) → GCF \(10x\): \(\;10x(2y + 3)\)
10. \(20x^{2}y + 30x\) → GCF \(10x\): \(\;10x(2xy + 3)\)
11. \(45x^{2}y - 15xy\) → GCF \(15xy\): \(\;15xy(3x - 1)\)
12. \(7x^{2}y + 21xy^{2}\) → GCF \(7xy\): \(\;7xy(x + 3y)\)

CautionCommon factor that is a bracket (click for solutions)

13. \(3(x + 2) + 5(x + 2)\) → GCF \((x + 2)\): \(\;8(x + 2)\)
14. \(2x(3x - 5) + 4(3x - 5)\) → GCF \((3x - 5)\): \(\;2(3x - 5)(x + 2)\)
15. \(6(x - 3) - 5(-x + 3)\) → GCF \((x - 3)\): \(\;11(x - 3)\)
16. \((3x + 2)(2x - 5) + (4x - 1)(2x - 5)\) → GCF \((2x - 5)\): \(\;(2x - 5)(7x + 1)\)

CautionGrouping — the GCF is \((x + 3)\) (click for solutions)

17. \(2x^{2} + 6x + x + 3 = 2x(x + 3) + (x + 3)\) → \((x + 3)(2x + 1)\)
18. \(3x^{2} + 9x + 2x + 6 = 3x(x + 3) + 2(x + 3)\) → \((x + 3)(3x + 2)\)

3.3 Expanding

Expanding is the inverse operation of factoring: it turns an expression written as a product into a sum of terms, generally in polynomial form.¹ It relies on distributivity, as in the rule \(a(b + c) = ab + ac\). For example, expanding \(3(x + 4)\) gives \(3x + 12\), while \((x + 2)(x - 5)\) expands to \(x^{2} - 3x - 10\).

Expanding lets us simplify or compare expressions, prepare an equation for solving, or reveal the standard form of a polynomial.

Worked example — a cost function

\[ C(x) = 5x(x + 20)(y + 3) \]

1. Simple distributivity: \(5x(x + 20) = 5x^{2} + 100x\).
2. Double distributivity: \((5x^{2} + 100x)(y + 3) = 5x^{2}y + 15x^{2} + 100xy + 300x\).

We obtain \(C(x) = 5x^{2}y + 100xy + 15x^{2} + 300x\), made up of increasing variable costs (\(5x^{2}y + 15x^{2}\)) and proportional variable costs (\(100xy + 300x\)).

CautionExercises — expand (click for solutions)

1. \(3(x + 4)\) → \(3x + 12\)
2. \(-2(x - 7)\) → \(-2x + 14\)
3. \(-4(x^{2} - 3x + 7)\) → \(-4x^{2} + 12x - 28\)
4. \(-4x(x^{2} - 3x + 7)\) → \(-4x^{3} + 12x^{2} - 28x\)
5. \((x + 2)(x + 3)\) → \(x^{2} + 5x + 6\)
6. \((2x - 5)(x^{2} + x + 1)\) → \(2x^{3} - 3x^{2} - 3x - 5\)
7. \((x + 1)(x - 2)(x + 3)\) → \(x^{3} + 2x^{2} - 5x - 6\)

3.4 Notable Identities

Notable identities are fundamental algebraic equalities that let us expand or factor certain expressions quickly, without a term-by-term calculation. The most common are:

\[ (a + b)^{2} = a^{2} + 2ab + b^{2}, \quad (a - b)^{2} = a^{2} - 2ab + b^{2}, \quad (a - b)(a + b) = a^{2} - b^{2}. \]

For instance, \(x^{2} + 6x + 9\) is immediately recognised as \((x + 3)^{2}\).

CautionDerivation — the square of a sum (click to expand)

Starting from the definition of a square and using distributivity,

\[ (a + b)^{2} = (a + b)(a + b) = a\,a + a\,b + b\,a + b\,b = a^{2} + ab + ab + b^{2} = a^{2} + 2ab + b^{2}. \]

The middle term \(2ab\) comes from the two identical cross-products \(ab\) and \(ba\). Replacing \(b\) by \(-b\) gives \((a - b)^{2} = a^{2} - 2ab + b^{2}\), and keeping only the cross-products that survive gives \((a - b)(a + b) = a^{2} - b^{2}\).

CautionExpand with notable identities (click for solutions)

1. \((x + 5)^{2}\) → \(x^{2} + 10x + 25\)
2. \((2x - 5)^{2}\) → \(4x^{2} - 20x + 25\)
3. \((-x + 5)^{2}\) → \(x^{2} - 10x + 25\)
4. \((2x - 5)(2x + 5)\) → \(4x^{2} - 25\)

CautionFactor with notable identities (click for solutions)

1. \(x^{2} + 6x + 9\) → \((x + 3)^{2}\)
2. \(9x^{2} - 24x + 16\) → \((3x - 4)^{2}\)
3. \(-4x^{2} + 12x - 9\) → \(-(2x - 3)^{2}\)
4. \(16x^{2} - 25\) → \((4x + 5)(4x - 5)\)
5. \(49x^{2} - 1\) → \((7x + 1)(7x - 1)\)
6. \(49x^{4} - x^{2}\) → \(x^{2}(7x + 1)(7x - 1)\)
7. \((x + 2)^{2} - (x - 2)^{2}\) → \(8x\)

3.5 Solving First-Degree Equations

TipGeneral method — first-degree equation \(ax + b = 0\)

1. Expand / remove parentheses if needed.
2. Group the terms in \(x\) on one side and the constants on the other.
3. Isolate \(x\): \(\;x = -\dfrac{b}{a}\) if \(a \neq 0\).

ImportantExam tip

If \(a = 0\): the equation is either impossible (no solution) when \(b \neq 0\), or an identity (infinitely many solutions) when \(b = 0\). Always check the coefficient of \(x\) before dividing.

Worked examples

1. \(3x - 7 = 11 \Rightarrow 3x = 18 \Rightarrow x = 6\).
2. \(2(x - 3) = 5 - x \Rightarrow 2x - 6 = 5 - x \Rightarrow 3x = 11 \Rightarrow x = \dfrac{11}{3}\).
3. \(\dfrac{x - 1}{3} + \dfrac{2x + 5}{6} = 2\): multiply by \(6\) to get \(2(x - 1) + (2x + 5) = 12\), i.e. \(4x + 3 = 12\), so \(x = \dfrac{9}{4}\).

3.6 Break-Even Formula

TipBreak-even quantity

\[ Q^* = \frac{F}{p - v} \]

where \(F\) = fixed cost, \(p\) = selling price, and \(v\) = variable cost per unit. The break-even quantity is the number of units where revenue exactly covers total cost.

CautionMixed exercises (click for solutions)

1. Simplify \(\dfrac{6x - 9}{3}\) → \(2x - 3\)
2. Factor \(4x^{2} - 12x\) → \(4x(x - 3)\)
3. Expand \((3x - 2)(x + 5)\) → \(3x^{2} + 13x - 10\)
4. Solve \(5x + 7 = 2x - 5\) → \(x = -4\)

4 Worked Example

4.1 Worked example — break-even for a café

Using the Opening Problem values: \(F = 2{,}000\), \(v = 3\), \(p = 8\).

Revenue function:

\[R(q) = 8q\]

Total cost function:

\[C(q) = 2{,}000 + 3q\]

Break-even: set \(R = C\):

\[ 8q = 2{,}000 + 3q \Rightarrow 5q = 2{,}000 \Rightarrow q^* = 400 \text{ units.} \]

Verification: \(R(400) = 8 \times 400 = 3{,}200\) and \(C(400) = 2{,}000 + 3 \times 400 = 3{,}200\). ✓

5 Guided Practice

A freelance designer charges €50 per hour. Her monthly fixed costs (software, office) total €600. How many hours must she bill to break even?

TipHint

Set up \(R(h) = 50h\) and \(C(h) = 600\). Then solve \(R(h) = C(h)\).

CautionShow the solution

Revenue function: \(R(h) = 50h\)

Total cost function: \(C(h) = 600\)

Break-even: \(50h = 600 \Rightarrow h = 12\) hours.

Verification: \(R(12) = C(12) = 600\). ✓

6 Interactive Exploration

Use the sliders below to explore how fixed cost, variable cost, and selling price interact to determine the break-even quantity.

viewof fixedCost = Inputs.range([0, 5000], {
  value: 2000,
  step: 100,
  label: "Fixed cost F (€)"
})

viewof variableCost = Inputs.range([0, 100], {
  value: 3,
  step: 0.5,
  label: "Variable cost per unit v (€)"
})

viewof price = Inputs.range([0, 200], {
  value: 8,
  step: 0.5,
  label: "Selling price per unit p (€)"
})

margin = price - variableCost
bep = margin > 0 ? fixedCost / margin : Infinity
revenueAtBep = price * bep
costAtBep = fixedCost + variableCost * bep
profitAtBep = revenueAtBep - costAtBep

html`
<div class="metric-grid">
  <div class="metric-card">
    <span class="metric-label">Contribution margin (p − v)</span>
    <strong>${margin.toFixed(2)} €</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">Break-even quantity Q*</span>
    <strong>${margin > 0 ? bep.toFixed(0) + " units" : "No break-even (p ≤ v)"}</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">Revenue at Q*</span>
    <strong>${margin > 0 ? revenueAtBep.toFixed(2) + " €" : "—"}</strong>
  </div>
  <div class="metric-card">
    <span class="metric-label">Total cost at Q*</span>
    <strong>${margin > 0 ? costAtBep.toFixed(2) + " €" : "—"}</strong>
  </div>
</div>
`

6.1 Investigation

1. Set the selling price equal to the variable cost. What happens to the break-even quantity? Why?
2. Double the fixed cost. How does \(Q^*\) change? Is the relationship linear?
3. A competitor offers the same product at €6. What break-even quantity would your café need at this price?

7 Applied Problem

A technology startup developing a mobile application estimates: fixed monthly costs of €8,500 (salaries, hosting, licences); variable cost of €0.50 per active user per month; subscription price of €4.99 per user per month.

1. Write the revenue function \(R(u)\) and the total cost function \(C(u)\), where \(u\) is the number of active users.
2. Calculate the break-even number of users.
3. If the startup currently has 1,800 users, is it profitable? By how much?
4. What selling price would reduce the break-even point to 2,000 users, keeping all other parameters fixed?

8 Common Errors

WarningCommon error — distributing a factor incorrectly

A student writes \(3(x + 4) = 3x + 4\).

The correct calculation is:

\[3(x + 4) = 3x + 12.\]

The factor 3 must multiply every term inside the parentheses. Omitting to multiply 4 by 3 is the most frequent expansion error.

9 Check Your Understanding

1. Factor \(6x^2 - 9x\).
2. Why does the break-even formula require \(p > v\)? What does it mean economically if \(p \leq v\)?
3. A restaurant has \(F =\) €3,000/month, \(v =\) €4/cover, \(p =\) €18/cover. Calculate the break-even number of covers.
4. A student solves \(2(x - 3) = x + 4\) and gets \(x = -2\). Is this correct? Identify any error.
5. Expand \((2x - 5)^2\) using a notable identity.

10 Summary

1. Algebraic transformations (factoring, expanding) change the form of an expression without changing its value — they reveal structure that makes equations easier to solve.
2. The break-even quantity \(Q^* = F / (p - v)\) is the point where revenue equals total cost; it only exists when the selling price exceeds the variable cost per unit.
3. Solving a first-degree equation means isolating the unknown by performing identical operations on both sides — always verify the solution by substituting back.

11 Independent Work

• Complete factoring exercises 1–15 in the exercise bank.

• Complete expanding exercises 1–10.

• Solve break-even problems A–D.

• Record any errors in your error log.

• Review the glossary entries for: factoring, expanding, break-even point, contribution margin.

  12 Module 2 — Algebra and Equations

  Distributive law (expanding)

  \[ a(b + c) = ab + ac \]

  Every term inside the parentheses must be multiplied by the factor outside.

  ---

  Break-even quantity

  \[ Q^* = \frac{F}{p - v} \]

  \(F\) = fixed cost, \(p\) = selling price per unit, \(v\) = variable cost per unit. Requires \(p > v\) (positive contribution margin).

Footnotes

1. An expression formed only from products and sums of constants and variables (usually written \(x\), \(y\), \(z\), …).