Course Cheatsheet

Section 02: Equations & Problem-Solving Strategies

The IDEA Method

A systematic approach for solving word problems:

Identify: What type of problem are we solving?
Develop: Create a plan using appropriate methods
Execute: Carry out the solution carefully
Assess: Check your answer makes sense

Translating Words to Mathematics

English Phrase	Symbol	Example
“is”, “equals”	=	“The cost is €50” → \(C = 50\)
“less than”	<	“x is less than 10” → \(x < 10\)
“at least”	≥	“at least 5 units” → \(x ≥ 5\)
“at most”	≤	“at most 100” → \(x ≤ 100\)
“increased by”	+	“price increased by €5” → \(p + 5\)
“decreased by”	-	“reduced by 20%” → \(x - 0.2x\)
“of”, “times”	×	“30% of sales” → \(0.3S\)
“per”	÷	“cost per unit” → \(\frac{\text{total cost}}{\text{units}}\)

Business Vocabulary

Essential Terms:

Revenue (R): Total income = Price × Quantity
Cost (C): Fixed costs + Variable costs
Profit (P): Revenue - Cost = R - C
Break-even: When Revenue = Cost (Profit = 0)
Margin: Profit as percentage of revenue
Markup: Increase from cost to selling price

Linear Equations

Standard Form: \(ax + b = c\)

Solving Multi-Step Equations:

Clear fractions by multiplying by LCD
Expand parentheses using distributive property
Collect like terms (variables on one side, constants on other)
Isolate variable by dividing by coefficient
Verify by substituting back

Example with Fractions: \[\frac{2x - 1}{3} + \frac{x + 2}{4} = 5\]

LCD = 12
Multiply through: \(4(2x - 1) + 3(x + 2) = 60\)
Expand: \(8x - 4 + 3x + 6 = 60\)
Solve: \(11x = 58\), so \(x = \frac{58}{11}\)

Inequalities

Key Rule

When multiplying or dividing by a negative number, flip the inequality sign!

Example: \(-2x > 6\)

Divide by -2: \(x < -3\) (sign flipped!)

Solution Notation:

\(x < a\): interval \((-\infty, a)\)
\(x ≤ a\): interval \((-\infty, a]\)
\(x > a\): interval \((a, \infty)\)
\(a < x < b\): interval \((a, b)\)

Systems of Linear Equations

2×2 Systems

Two Methods:

1. Substitution Method (best when one variable is isolated):

Isolate one variable in one equation
Substitute into the other equation
Solve for remaining variable
Back-substitute to find first variable

2. Elimination Method (best for symmetric systems):

Align equations vertically
Multiply to create opposite coefficients
Add/subtract to eliminate one variable
Solve for remaining variable

Three Possible Outcomes:

Unique Solution: Lines intersect once (most common)
No Solution: Parallel lines (inconsistent system)
Infinite Solutions: Same line (dependent equations)

Quick Classification

For system \(\begin{cases} a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2 \end{cases}\):

If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} ≠ \frac{c_1}{c_2}\): No solution
If \(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\): Infinite solutions
Otherwise: Unique solution

Quadratic Equations

Standard Form: \(ax^2 + bx + c = 0\)

The Discriminant

\(\Delta = b^2 - 4ac\) tells us:

\(\Delta\) Value	Solution Type	Graph Behavior
\(\Delta > 0\) (perfect square)	Two rational solutions	Crosses x-axis twice
\(\Delta > 0\) (not perfect square)	Two irrational solutions	Crosses x-axis twice
\(\Delta = 0\)	One repeated solution	Touches x-axis once
\(\Delta < 0\)	No real solutions	Doesn’t touch x-axis

Three Solution Methods

1. Factoring (when \(\Delta\) is a perfect square):

Factor the quadratic
Apply Zero Product Property: If \(AB = 0\), then \(A = 0\) or \(B = 0\)

2. Quadratic Formula (always works):

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

3. Completing the Square (useful for deriving vertex form):

Move constant to right side
Add \((\frac{b}{2a})^2\) to both sides
Factor left side as perfect square

Method Selection Guide

Calculate Δ = b² - 4ac
│
├─ Δ < 0 → No real solutions
├─ Δ = 0 → One solution: x = -b/(2a)
└─ Δ > 0 → Two real solutions
           └─ Is Δ a perfect square?
              ├─ YES → Try factoring first
              └─ NO → Use quadratic formula

Biquadratic Equations

Form: \(ax^4 + bx^2 + c = 0\)

Solution Strategy:

Let \(u = x^2\)
Solve \(au^2 + bu + c = 0\) (quadratic in \(u\))
Back-substitute: If \(u = k\), then \(x^2 = k\)
Solve for \(x\): \(x = \pm\sqrt{k}\) (if \(k ≥ 0\))

Example: \(x^4 - 5x^2 + 4 = 0\)

Let \(u = x^2\): \(u^2 - 5u + 4 = 0\)
Factor: \((u - 1)(u - 4) = 0\)
So \(u = 1\) or \(u = 4\)
Therefore: \(x = \pm 1\) or \(x = \pm 2\)

Fractional (Rational) Equations

Key Steps:

Find domain restrictions (denominators ≠ 0)
Clear fractions by multiplying by LCD
Solve resulting equation
Check solutions against domain restrictions

Example: \(\frac{2}{x-1} + \frac{3}{x+2} = 1\)

Domain: \(x ≠ 1, x ≠ -2\)
LCD: \((x-1)(x+2)\)
Clear fractions: \(2(x+2) + 3(x-1) = (x-1)(x+2)\)
Expand and solve: \(x^2 - 4x - 5 = 0\)
Solutions must be checked against domain!

Cross Multiplication: For \(\frac{a}{b} = \frac{c}{d}\), we get \(ad = bc\)

Radical Equations

Solution Strategy:

Isolate the radical term
Square both sides (or raise to appropriate power)
Solve resulting equation
CHECK ALL SOLUTIONS

Really, don’t forget checking the solutions

Squaring can introduce extraneous solutions!

Example: \(\sqrt{x + 3} = x - 1\)

Square both sides: \(x + 3 = (x - 1)^2\)
Expand: \(x + 3 = x^2 - 2x + 1\)
Rearrange: \(x^2 - 3x - 2 = 0\)
Solutions must be checked in original equation!

Multiple Radicals:

Isolate one radical at a time
Square, simplify, repeat if necessary

Cubic Equations (Recap from Section 01)

Form: \(ax^3 + bx^2 + cx + d = 0\)

Solution Strategies:

1. Rational Root Theorem: Possible rational roots = \(\pm\frac{\text{factors of } d}{\text{factors of } a}\)

2. Special Forms:

Sum of cubes: \(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)
Difference of cubes: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

3. Factor by Grouping:

Look for common factors in pairs of terms

Example: \(x^3 - 7x^2 + 14x - 8 = 0\)

Test rational roots: Try \(x = 1\): \(1 - 7 + 14 - 8 = 0\) ✓
Factor out \((x - 1)\) using synthetic division
Get: \((x - 1)(x^2 - 6x + 8) = 0\)
Factor further: \((x - 1)(x - 2)(x - 4) = 0\)

Exponential Equations

Basic Strategies:

1. Same Base Method:

If \(a^f(x) = a^{g(x)}\), then \(f(x) = g(x)\)

2. Logarithm Method:

For \(a^x = b\), take log of both sides:

\(\log(a^x) = \log(b)\)
\(x\log(a) = \log(b)\)
\(x = \frac{\log(b)}{\log(a)}\)

3. Substitution for Complex Forms:

For \(4^x - 3 \cdot 2^x + 2 = 0\):

Note: \(4^x = (2^2)^x = (2^x)^2\)
Let \(u = 2^x\): \(u^2 - 3u + 2 = 0\)

Mixed Base Systems:

For different bases, use logarithms strategically or look for relationships

Logarithmic Equations

Key Strategies:

1. Use Properties to Combine:

Product: \(\log_a(x) + \log_a(y) = \log_a(xy)\)
Quotient: \(\log_a(x) - \log_a(y) = \log_a(\frac{x}{y})\)
Power: \(n\log_a(x) = \log_a(x^n)\)

2. Convert to Exponential Form:

If \(\log_a(x) = y\), then \(a^y = x\)

3. Domain Restrictions:

Always ensure arguments of logarithms are positive!

Example: \(\log(x) + \log(x - 3) = 1\)

Domain: \(x > 3\)
Combine: \(\log(x(x - 3)) = 1\)
Convert: \(x(x - 3) = 10\)
Solve: \(x^2 - 3x - 10 = 0\)
Check domain restrictions!