What is the quadratic formula and when do I use it?

The quadratic formula x = (-b ± √(b² - 4ac))/(2a) solves any quadratic equation in the form ax² + bx + c = 0. Use it when factoring is difficult or impossible, or when you need exact solutions. The discriminant (b² - 4ac) tells you whether solutions are real or complex.

How do I calculate the distance between two points?

Use the distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]. This is derived from the Pythagorean theorem and calculates the straight-line distance between points (x₁, y₁) and (x₂, y₂) in the coordinate plane.

What are the basic trigonometric ratios?

The three basic ratios are: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. Remember SOH-CAH-TOA as a mnemonic. These ratios relate angles to side lengths in right triangles.

How do I simplify expressions with exponents?

Apply exponent laws: aᵐ × aⁿ = aᵐ⁺ⁿ (multiply: add exponents), aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide: subtract exponents), (aᵐ)ⁿ = aᵐⁿ (power of power: multiply exponents). Remember a⁰ = 1 and a⁻ⁿ = 1/aⁿ.

What is the difference between area and perimeter?

Perimeter is the distance around the outside of a 2D shape (measured in length units like cm or m). Area is the space inside a 2D shape (measured in square units like cm² or m²). For a rectangle: perimeter = 2(l + w), area = l × w.

Prior Learning Formulae AI HL: Essential Foundation for IB Math Higher Level

Welcome to the comprehensive guide for Prior Learning Formulae in IB Mathematics Applications and Interpretation Higher Level. Before diving into advanced AI HL topics, establishing a rock-solid foundation in fundamental mathematics is absolutely essential. This guide covers all prerequisite concepts including algebra, geometry, coordinate geometry, trigonometry, and essential mathematical techniques that form the building blocks for success in IB Math AI HL. Mastering these foundational formulas ensures you can confidently tackle the more complex concepts that characterize Higher Level mathematics.

Why Prior Learning Matters for AI HL Success

IB Math AI HL builds upon years of mathematical learning, assuming fluency with fundamental concepts. Without solid prior knowledge, even brilliant students struggle with advanced topics. Prior learning formulas appear repeatedly throughout the AI HL curriculum—in calculus, statistics, modeling, and problem-solving. Understanding where these formulas come from and how to apply them efficiently creates a competitive advantage that separates top performers from those who merely survive the course.

Algebra Fundamentals

Algebra forms the language of mathematics, providing tools for generalizing patterns, solving equations, and manipulating expressions. Strong algebraic skills are non-negotiable for AI HL success, as virtually every advanced topic requires algebraic manipulation.

Quadratic Equations

Quadratic equations appear throughout IB Math AI HL in optimization problems, function analysis, and modeling scenarios. The ability to solve quadratics quickly and accurately is fundamental. Explore our comprehensive Quadratic Equation Calculator for practice.

The Quadratic Formula

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

Solves \( ax^2 + bx + c = 0 \) where \( a \neq 0 \)

The Discriminant

\[ \Delta = b^2 - 4ac \]

Determines nature of roots:

\( \Delta > 0 \): Two distinct real roots
\( \Delta = 0 \): One repeated real root (vertex touches x-axis)
\( \Delta < 0 \): Two complex conjugate roots (no real solutions)

Example: Solving a Quadratic

Solve \( 2x^2 - 5x - 3 = 0 \)

Solution: \( a = 2 \), \( b = -5 \), \( c = -3 \)

\( x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)} \)

\( x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4} \)

\( x = \frac{12}{4} = 3 \) or \( x = \frac{-2}{4} = -\frac{1}{2} \)

Factoring Techniques

Difference of Squares

\[ a^2 - b^2 = (a+b)(a-b) \]

Perfect Square Trinomials

\[ a^2 + 2ab + b^2 = (a+b)^2 \] \[ a^2 - 2ab + b^2 = (a-b)^2 \]

Sum and Difference of Cubes

\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \] \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]

Laws of Exponents

Exponent laws are essential for simplifying expressions, solving exponential equations, and understanding exponential functions—a major AI HL topic.

Fundamental Exponent Laws

\[ a^m \times a^n = a^{m+n} \quad \text{(multiply: add exponents)} \] \[ \frac{a^m}{a^n} = a^{m-n} \quad \text{(divide: subtract exponents)} \] \[ (a^m)^n = a^{mn} \quad \text{(power of power: multiply)} \] \[ (ab)^n = a^n b^n \quad \text{(distribute exponent)} \] \[ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \]

Special Exponent Cases

\[ a^0 = 1 \quad \text{(for } a \neq 0\text{)} \] \[ a^{-n} = \frac{1}{a^n} \quad \text{(negative exponent)} \] \[ a^{\frac{1}{n}} = \sqrt[n]{a} \quad \text{(fractional exponent)} \] \[ a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \]

Geometry Formulas

Geometric formulas provide essential tools for calculating areas, perimeters, volumes, and surface areas. These concepts appear in optimization problems, modeling tasks, and applied mathematics throughout AI HL.

2D Shapes - Area and Perimeter

Shape	Area Formula	Perimeter Formula
Rectangle	\( A = l \times w \)	\( P = 2(l + w) \)
Square	\( A = s^2 \)	\( P = 4s \)
Triangle	\( A = \frac{1}{2}bh \)	\( P = a + b + c \)
Circle	\( A = \pi r^2 \)	\( C = 2\pi r = \pi d \)
Trapezoid	\( A = \frac{1}{2}(b_1 + b_2)h \)	Sum of all sides
Parallelogram	\( A = bh \)	\( P = 2(a + b) \)

Understanding Area vs Perimeter

Perimeter: The distance around the outside of a shape (linear measurement in cm, m, etc.)

Area: The amount of space inside a shape (square measurement in cm², m², etc.)

Key Insight: You can have shapes with the same perimeter but different areas, and vice versa!

3D Shapes - Volume and Surface Area

Shape	Volume Formula	Surface Area Formula
Rectangular Prism	\( V = l \times w \times h \)	\( SA = 2(lw + lh + wh) \)
Cube	\( V = s^3 \)	\( SA = 6s^2 \)
Cylinder	\( V = \pi r^2 h \)	\( SA = 2\pi r^2 + 2\pi rh \)
Sphere	\( V = \frac{4}{3}\pi r^3 \)	\( SA = 4\pi r^2 \)
Cone	\( V = \frac{1}{3}\pi r^2 h \)	\( SA = \pi r^2 + \pi r l \)*
Pyramid	\( V = \frac{1}{3}Bh \)**	\( SA = B + \text{lateral area} \)

* \( l \) = slant height, ** \( B \) = base area

Volume Pattern Recognition

Notice the \( \frac{1}{3} \) factor in cone and pyramid volumes? This reflects that cones and pyramids are "pointed" versions of cylinders and prisms, occupying exactly one-third the volume of their corresponding prism/cylinder with the same base and height.

Coordinate Geometry

Coordinate geometry connects algebra and geometry, allowing us to analyze geometric shapes using algebraic methods. These formulas are fundamental for understanding functions, optimization, and analytical geometry in AI HL.

Distance and Midpoint

Distance Formula

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Calculates distance between points \( (x_1, y_1) \) and \( (x_2, y_2) \)

Derived from Pythagorean theorem

Midpoint Formula

\[ M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \]

Finds the point exactly halfway between two given points

Example: Distance and Midpoint

Find the distance and midpoint between \( A(2, 3) \) and \( B(8, 11) \)

Distance:

\( d = \sqrt{(8-2)^2 + (11-3)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \) units

Midpoint:

\( M = \left(\frac{2+8}{2}, \frac{3+11}{2}\right) = (5, 7) \)

Slope and Linear Equations

Slope Formula

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\text{rise}}{\text{run}} \]

Measures steepness and direction of a line

Slope-Intercept Form

\[ y = mx + c \]

where \( m \) = slope, \( c \) = y-intercept

Point-Slope Form

\[ y - y_1 = m(x - x_1) \]

Useful when you know slope and one point

Parallel and Perpendicular Lines

Parallel lines: Have equal slopes (\( m_1 = m_2 \))

Perpendicular lines: Have slopes whose product is \( -1 \) (\( m_1 \times m_2 = -1 \))

Example: If one line has slope \( 3 \), a perpendicular line has slope \( -\frac{1}{3} \)

Trigonometry Fundamentals

Trigonometry connects angles with ratios of sides in triangles. These relationships extend to circular functions, wave modeling, and periodic phenomena—all critical for AI HL success.

Right Triangle Trigonometry

Basic Trigonometric Ratios (SOH-CAH-TOA)

\[ \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \] \[ \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] \[ \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta} \]

Reciprocal Trigonometric Functions

\[ \csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}} \] \[ \sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}} \] \[ \cot\theta = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}} \]

Pythagorean Theorem

\[ a^2 + b^2 = c^2 \]

For right triangles: \( a \) and \( b \) are legs, \( c \) is hypotenuse

Pythagorean Identities

\[ \sin^2\theta + \cos^2\theta = 1 \] \[ 1 + \tan^2\theta = \sec^2\theta \] \[ 1 + \cot^2\theta = \csc^2\theta \]

Special Angle Values

Angle	sin	cos	tan
0°	0	1	0
30° (π/6)	1/2	√3/2	1/√3
45° (π/4)	√2/2	√2/2	1
60° (π/3)	√3/2	1/2	√3
90° (π/2)	1	0	undefined

Memorization Tip for Special Angles

For sine values at 0°, 30°, 45°, 60°, 90°, remember the pattern:

\( \sin\theta = \frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2} \)

Cosine values are the same pattern in reverse!

Radicals and Rational Expressions

Simplifying Radicals

Radical Rules

\[ \sqrt{ab} = \sqrt{a} \times \sqrt{b} \] \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \] \[ \sqrt[n]{a^m} = a^{\frac{m}{n}} \] \[ \sqrt{a} \times \sqrt{a} = a \]

Rationalizing the Denominator

\[ \frac{1}{\sqrt{a}} = \frac{1}{\sqrt{a}} \times \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} \] \[ \frac{1}{a + \sqrt{b}} = \frac{1}{a + \sqrt{b}} \times \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b} \]

Percentages and Proportions

Percentage calculations appear throughout statistics, financial mathematics, and data analysis in AI HL. Master these fundamentals to handle more complex percentage problems confidently. Practice with our Percentage Calculator.

Percentage Formulas

\[ \text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\% \] \[ \text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole} \] \[ \text{Percentage Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\% \]

Interactive Quadratic Equation Solver

Quadratic Equation Solver

Solve quadratic equations in the form \( ax^2 + bx + c = 0 \)

Coefficient a: Coefficient b: Coefficient c:

Study Strategies for Prior Learning

Building Strong Foundations

Regular Review: Dedicate 15-20 minutes daily to reviewing prior learning formulas. Spaced repetition builds long-term retention
Active Practice: Don't just read formulas—work through problems using each formula multiple times
Create Formula Sheets: Handwrite your own formula reference sheets. The act of writing reinforces memory
Understand Derivations: Know where formulas come from. Understanding beats memorization for complex applications
Connect Concepts: Recognize how formulas relate to each other (e.g., distance formula from Pythagorean theorem)
Use Technology Wisely: Verify hand calculations with calculators, but always work problems by hand first

Common Mistakes to Avoid

Common Error	Correct Approach	Example
Sign errors in quadratic formula	Carefully apply \( -b \) (negative of b)	If \( b = -5 \), then \( -b = 5 \)
Forgetting to square both terms in distance formula	Square \( (x_2 - x_1) \) AND \( (y_2 - y_1) \)	\( d = \sqrt{(\Delta x)^2 + (\Delta y)^2} \)
Mixing up area and perimeter formulas	Remember units: area is squared	Rectangle: Area = \( lw \), Perimeter = \( 2(l+w) \)
Incorrectly applying exponent laws	Multiply exponents only for power of power	\( (x^2)^3 = x^6 \), but \( x^2 \times x^3 = x^5 \)
Confusing sin and cos	Use SOH-CAH-TOA systematically	sin is opposite/hypotenuse, cos is adjacent/hypotenuse

Connecting Prior Learning to AI HL Topics

Understanding how prior learning connects to advanced AI HL content helps you recognize when to apply these foundational formulas.

Where Prior Learning Appears in AI HL

Quadratics in Optimization: Maximum/minimum problems in calculus require completing the square and finding vertices
Geometry in Calculus: Volume and surface area formulas extend to calculus applications (rates of change, related rates)
Coordinate Geometry in Functions: Distance and midpoint formulas analyze function transformations and composite functions
Trigonometry in Modeling: Sine and cosine functions model periodic phenomena (waves, seasons, oscillations)
Exponents in Financial Math: Compound interest and exponential growth require fluent exponent manipulation
Algebra in Statistics: Regression analysis and correlation require algebraic manipulation of formulas

Assessment and Self-Testing

Prior Knowledge Checklist

Can You Do These Without Looking?

✓ Solve any quadratic equation using the quadratic formula
✓ Calculate distance and midpoint between two coordinate points
✓ Find slope of a line from two points and write equation in slope-intercept form
✓ Apply Pythagorean theorem to find missing side of right triangle
✓ Calculate sine, cosine, and tangent of angles in right triangles
✓ Simplify expressions using exponent laws
✓ Calculate area and perimeter of common 2D shapes
✓ Calculate volume and surface area of common 3D shapes
✓ Rationalize denominators containing radicals
✓ Calculate percentage change and percentage of amounts

Additional Resources

Strengthen your prior learning foundation with these RevisionTown resources:

Functions Formulae AI SL & AI HL - Connect prior learning to function concepts
Number and Algebra Formulae AI SL & AI HL - Build on algebraic foundations
Quadratic Equation Calculator - Practice solving quadratics
Percentage Calculator - Master percentage calculations
Triangle Area Calculator - Geometry practice
Circle Area Calculator - More geometry tools
Sphere Volume Calculator - 3D geometry practice
Distance Calculator - Coordinate geometry tool
IB Mathematics AA vs AI Comparison - Understand your course
IB Diploma Points Calculator - Track your IB progress

GDC Calculator Skills for Prior Learning

Essential Calculator Functions

Quadratic Solver: Learn equation solver mode to verify quadratic solutions
Graphing: Visualize equations to understand solution behavior
Table Mode: Generate value tables to identify patterns
Trigonometric Functions: Ensure calculator is in correct mode (degrees vs radians)
Fractional Display: Learn to display exact answers as fractions when appropriate
Memory Functions: Store intermediate values to avoid rounding errors

Preparing for AI HL Success

Prior learning is not just "review"—it's the foundation upon which all IB Math AI HL concepts are built. Students who struggle with AI HL often lack fluency with these fundamental formulas, not because they lack intelligence, but because they haven't invested time in building automatic recall and application skills.

The 80/20 Rule for Prior Learning

Approximately 80% of AI HL problem-solving relies on 20% of prior learning concepts. Focus intensely on:

Algebraic manipulation (solving equations, simplifying expressions)
Quadratic equations (appear in calculus, optimization, modeling)
Coordinate geometry (foundations for functions and calculus)
Basic trigonometry (extends to circular functions and modeling)
Exponent rules (essential for exponential and logarithmic functions)

Time Investment Strategy

Recommended Study Schedule

Week 1-2: Algebra fundamentals (3 hours total)

Quadratic equations and discriminant
Factoring techniques
Exponent and radical laws

Week 3-4: Geometry (2 hours total)

Area and perimeter formulas
Volume and surface area
Practice mixed problems

Week 5-6: Coordinate geometry and trigonometry (3 hours total)

Distance, midpoint, slope
Trigonometric ratios
Special angle values

Ongoing: 15 minutes daily review and practice

Conclusion

Mastering prior learning formulas is non-negotiable for IB Math AI HL success. These foundational concepts appear repeatedly throughout the curriculum, from basic problem-solving to advanced calculus and statistical applications. Students who invest time strengthening their prior knowledge create a competitive advantage that pays dividends throughout the two-year IB program and beyond.

Remember that mathematical fluency develops through consistent practice. Don't just memorize formulas—understand their derivations, recognize when to apply them, and practice until application becomes automatic. The confidence that comes from rock-solid foundations transforms challenging AI HL topics from overwhelming obstacles into manageable challenges.

Continue building your IB Mathematics foundation through our comprehensive collection of IB Mathematics resources, practice with interactive calculators, and connect prior learning to advanced topics throughout your AI HL journey. Success in IB Math AI HL begins with mastery of these essential prior learning formulas.