An equation is a mathematical statement that shows two expressions are equal, connected by an equals sign (=).

Think of an equation like a balanced scale - what's on the left side must equal what's on the right side.

Examples:

• \( x + 3 = 7 \) (the unknown \(x\) plus 3 equals 7)
• \( 2y = 10 \) (two times \(y\) equals 10)
• \( 5 + 2 = 7 \) (a true equation with no unknowns)

The goal is usually to find the value of the unknown variable that makes the equation true.

The simplest equations require only one operation to solve. These help you understand the basic principle of 'doing the same thing to both sides'.

Addition/Subtraction Examples:

• \( x + 5 = 12 \) → Subtract 5: \( x = 7 \)
• \( y - 3 = 8 \) → Add 3: \( y = 11 \)

Multiplication/Division Examples:

• \( 4x = 20 \) → Divide by 4: \( x = 5 \)
• \( \frac{z}{3} = 6 \) → Multiply by 3: \( z = 18 \)

Two-step equations require two operations to isolate the variable. Follow the order: undo addition/subtraction first, then multiplication/division.

Step-by-Step Process:

1. Identify the operations performed on the variable
2. Undo addition/subtraction first
3. Undo multiplication/division second

Example: \( 3x + 7 = 22 \)

• Step 1: Subtract 7 from both sides → \( 3x = 15 \)
• Step 2: Divide both sides by 3 → \( x = 5 \)

Check: \( 3(5) + 7 = 15 + 7 = 22 \) ✓

More complex linear equations may involve combining like terms, distributing, or variables on both sides.

Strategies:

1. Distribute if needed: \( 2(x + 3) = 2x + 6 \)
2. Combine like terms on each side
3. Move variables to one side, constants to the other
4. Solve the resulting two-step equation

Example: \( 4(x - 2) + 3 = 2x + 7 \)

• Distribute: \( 4x - 8 + 3 = 2x + 7 \)
• Combine like terms: \( 4x - 5 = 2x + 7 \)
• Subtract 2x: \( 2x - 5 = 7 \)
• Add 5: \( 2x = 12 \)
• Divide by 2: \( x = 6 \)

Quadratic equations have the standard form \( ax^2 + bx + c = 0 \) where \(a \neq 0\).

The highest power of the variable is 2, creating a parabola when graphed.

Key Characteristics:

• Degree: 2 (highest power)
• Graph: U-shaped curve (parabola)
• Solutions: Can have 0, 1, or 2 real solutions

Simplest Form: \( x^2 = k \)

• If \(k > 0\): Two solutions \(x = \pm\sqrt{k}\)
• If \(k = 0\): One solution \(x = 0\)
• If \(k < 0\): No real solutions

Example: \( x^2 = 16 \) → \( x = \pm 4 \)

Factoring is often the fastest method when the quadratic can be written as a product of two linear factors.

Zero Product Property: If \(ab = 0\), then \(a = 0\) or \(b = 0\) (or both).

Common Patterns:

• Difference of squares: \(x^2 - a^2 = (x-a)(x+a)\)
• Perfect square trinomial: \(x^2 ± 2ax + a^2 = (x ± a)^2\)
• General trinomial: \(x^2 + bx + c = (x + m)(x + n)\) where \(m + n = b\) and \(mn = c\)

Example: \( x^2 - 5x + 6 = 0 \)

• Find factors of 6 that add to -5: -2 and -3
• Factor: \( (x - 2)(x - 3) = 0 \)
• Solutions: \( x = 2 \) or \( x = 3 \)

The quadratic formula works for ANY quadratic equation, even when factoring is difficult or impossible.

For equation \(ax^2 + bx + c = 0\), identify \(a\), \(b\), and \(c\), then substitute.

The Discriminant: \(\Delta = b^2 - 4ac\)

• If \(\Delta > 0\): Two different real solutions
• If \(\Delta = 0\): One repeated real solution
• If \(\Delta < 0\): No real solutions (complex solutions)

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

• \(a = 2\), \(b = 3\), \(c = -2\)
• \(\Delta = 3^2 - 4(2)(-2) = 9 + 16 = 25\)
• \(x = \frac{-3 \pm \sqrt{25}}{2(2)} = \frac{-3 \pm 5}{4}\)
• Solutions: \(x = \frac{1}{2}\) or \(x = -2\)

A system of equations is a set of two or more equations with the same variables. The solution is the point(s) where all equations are satisfied simultaneously.

Methods for Solving:

1. Graphical Method: Plot both lines and find intersection
2. Substitution Method: Solve one equation for a variable, substitute into the other
3. Elimination Method: Add/subtract equations to eliminate a variable

Example System:

\( \begin{cases} x + y = 8 \\ 2x - y = 1 \end{cases} \)

Solution by Elimination:

• Add the equations: \((x + y) + (2x - y) = 8 + 1\)
• Simplify: \(3x = 9\)
• Solve: \(x = 3\)
• Substitute back: \(3 + y = 8\) → \(y = 5\)
• Solution: \((3, 5)\)

Word problems translate real-world scenarios into mathematical equations. The key is to identify the unknown quantities and the relationships between them.

Steps to solve:

• Read and Understand: Identify what you need to find.
• Define Variables: Assign letters (like \( p \)) to the unknown quantities.
• Formulate the Equation(s): Write equations that represent the problem.
• Solve the Equation(s): Use algebraic methods to find the value of the variables.
• Check Your Answer: Make sure the solution makes sense in the context of the problem.

Example: You buy a book and a pen for a total of \( \$15 \). The book costs \( \$9 \) more than the pen. What are the prices of the book and the pen?

Solution:

• Let \( p \) be the price of the pen.
• The price of the book is \( p + 9 \).
• Equation: \( p + (p + 9) = 15 \)
• Solve: \( 2p + 9 = 15 \) \( \Rightarrow \) \( 2p = 6 \) \( \Rightarrow \) \( p = 3 \).
• The pen costs \( \$3 \) and the book costs \( \$12 \).

Cubic equations have degree 3 with the form \( ax^3 + bx^2 + cx + d = 0 \).

They always have at least one real solution and can have up to 3 real solutions.

Common Solution Methods:

• Factor by grouping: When possible to group terms
• Rational Root Theorem: Test factors of \(d/a\)
• Synthetic division: Once one root is found

Example: \( x^3 - 6x^2 + 9x = 0 \)

• Factor out common term: \( x(x^2 - 6x + 9) = 0 \)
• Factor the quadratic: \( x(x - 3)^2 = 0 \)
• Solutions: \(x = 0\) and \(x = 3\) (with multiplicity 2)

Special Case - Difference/Sum of Cubes:

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

Real-world applications often involve systems of equations and polynomial modeling.

Common Application Areas:

• Physics: Projectile motion, wave equations
• Economics: Supply and demand, profit optimization
• Engineering: Structural design, signal processing
• Biology: Population growth, chemical reactions

Problem-Solving Strategy:

1. Identify the unknown variables
2. Write equations based on given constraints
3. Choose appropriate solution method
4. Verify solutions make physical sense

Example - Projectile Motion:

Height equation: \(h(t) = -16t^2 + 64t + 80\)

• When does projectile hit ground? Set \(h(t) = 0\)
• \(-16t^2 + 64t + 80 = 0\)
• Divide by -16: \(t^2 - 4t - 5 = 0\)
• Factor: \((t - 5)(t + 1) = 0\)
• Physical solution: \(t = 5\) seconds

Advanced equation systems combine multiple techniques and often appear in higher mathematics and engineering.

Types of Complex Systems:

• Nonlinear systems: Mixing linear and quadratic equations
• Three-variable systems: Requiring elimination in multiple steps
• Parametric equations: Where solutions depend on parameters
• Optimization problems: Finding maximum/minimum values

Example - Nonlinear System:

\( \begin{cases} x^2 + y^2 = 25 \\ x + y = 7 \end{cases} \)

Solution Strategy:

• From second equation: \(y = 7 - x\)
• Substitute: \(x^2 + (7-x)^2 = 25\)
• Expand: \(x^2 + 49 - 14x + x^2 = 25\)
• Simplify: \(2x^2 - 14x + 24 = 0\)
• Divide by 2: \(x^2 - 7x + 12 = 0\)
• Factor: \((x-3)(x-4) = 0\)
• Solutions: \((3,4)\) and \((4,3)\)

Level 1-3 Practice:

1. Solve: \( 3x + 7 = 22 \)
2. Solve: \( \frac{2y - 1}{3} = 5 \)
3. Solve: \( 4(z + 3) - 2z = 18 \)

Level 4-6 Practice:

4. Solve by factoring: \( x^2 - 7x + 12 = 0 \)
5. Use quadratic formula: \( 2x^2 + x - 3 = 0 \)
6. System of equations: \( \begin{cases} 2x + y = 9 \\ x - y = 3 \end{cases} \)

Level 7-10 Challenge:

7. Solve: \( x^3 - 8 = 0 \)
8. Word problem: A ball is thrown with height \( h(t) = -16t^2 + 48t + 64 \). When does it hit the ground?
9. Nonlinear system: \( \begin{cases} xy = 12 \\ x + y = 8 \end{cases} \)
10. Find all solutions to: \( x^4 - 10x^2 + 9 = 0 \)