2️⃣ Mastering Equations for Better Decision-Making

Abstract

Master algebraic expressions and equations using the essential tools of factoring, expanding, notable identities, and solving first-degree equations, in order to strengthen analytical and decision-support skills.

Learning objectives

• Simplify and rewrite an algebraic expression.
• Reveal a structure (common factor, notable identity).
• Solve first-degree equations in context.

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.

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 \(x = 3\).

Steps of factoring

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

Exercises — factor

Basic 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 \(7xy\): \(\;7xy(x + 3y)\)

Common 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 \((2x - 5)\): \(\;(2x - 5)(7x + 1)\)

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

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

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 (\(100xy + 300x\)).

Exercises — 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. \((x + 1)(x - 2)(x + 3)\) → \(x^{3} + 2x^{2} - 5x - 6\)

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 \((x + 3)^{2}\).

Derivation — 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 \((a - b)(a + b) = a^{2} - b^{2}\).

Expand 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. \((2x - 5)(2x + 5)\) → \(4x^{2} - 25\)

Factor 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. \((x + 2)^{2} - (x - 2)^{2}\) → \(8x\)

Solving First-Degree Equations

General 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 \(a \neq 0\).

Exam 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 \(x = \dfrac{9}{4}\).

Practice (mini-bank)

Mixed 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 \(5x + 7 = 2x - 5\) → \(x = -4\)

Footnotes

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