📖 What is Algebra?
Algebra is the branch of mathematics that uses letters (variables) to represent numbers in equations and formulas. It's the foundation of higher mathematics and is used everywhere from calculating interest rates to programming computers.
Why Learn Algebra?
- Problem Solving: Teaches logical thinking and systematic approaches to problems
- Real-World Applications: Used in science, engineering, finance, and technology
- Foundation for Advanced Math: Required for calculus, statistics, and physics
- Critical Thinking: Develops analytical skills transferable to any field
🎯 Key Concepts
1. Variables and Expressions
A variable is a letter that represents an unknown number. An algebraic expression combines variables, numbers, and operations.
Terms to Know
- Coefficient: The number in front of a variable (in $3x$, the coefficient is 3)
- Constant: A number without a variable (in $3x + 5$, the constant is 5)
- Like Terms: Terms with the same variable raised to the same power ($2x$ and $5x$ are like terms)
2. Linear Equations
A linear equation has variables raised only to the first power and graphs as a straight line. The standard form is $ax + b = c$.
where $m$ = slope and $b$ = y-intercept
where $(x_1, y_1)$ is a point on the line
3. Quadratic Equations
A quadratic equation has a variable raised to the second power. The standard form is $ax^2 + bx + c = 0$.
Solves any quadratic equation $ax^2 + bx + c = 0$
Methods for Solving Quadratics
- Factoring: Express as $(x + p)(x + q) = 0$ and solve each factor
- Quadratic Formula: Works for any quadratic (use when factoring is hard)
- Completing the Square: Rewrite in form $(x + k)^2 = n$
- Graphing: Find where the parabola crosses the x-axis
4. Systems of Equations
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 true simultaneously.
Methods for Solving Systems
- Substitution: Solve one equation for a variable, substitute into the other
- Elimination: Add/subtract equations to eliminate one variable
- Graphing: Graph both equations; the intersection is the solution
5. Polynomials
A polynomial is an expression with variables raised to whole number powers, combined using addition, subtraction, and multiplication.
Polynomial Operations
- Adding: Combine like terms
- Subtracting: Distribute the negative, then combine like terms
- Multiplying: Use FOIL for binomials, or distribute each term
- Factoring: Find common factors or use special patterns
📝 Essential Formulas
Quick Reference Table
| Formula Name | Formula | When to Use |
|---|---|---|
| Slope Formula | $m = \frac{y_2 - y_1}{x_2 - x_1}$ | Finding slope between two points |
| Distance Formula | $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ | Distance between two points |
| Midpoint Formula | $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$ | Finding point between two points |
| Quadratic Formula | $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ | Solving quadratic equations |
| Discriminant | $D = b^2 - 4ac$ | Determining number of solutions |
Special Factoring Patterns
Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product Rule | $a^m \cdot a^n = a^{m+n}$ | $x^2 \cdot x^3 = x^5$ |
| Quotient Rule | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{x^5}{x^2} = x^3$ |
| Power Rule | $(a^m)^n = a^{mn}$ | $(x^2)^3 = x^6$ |
| Zero Exponent | $a^0 = 1$ | $5^0 = 1$ |
| Negative Exponent | $a^{-n} = \frac{1}{a^n}$ | $x^{-2} = \frac{1}{x^2}$ |
✏️ Practice Problems
Test your understanding with these problems. Click "Show Solution" to see step-by-step explanations.
$3x + 7 - 7 = 22 - 7$
$3x = 15$
$\frac{3x}{3} = \frac{15}{3}$
$x = 5$
Answer: $x = 5$
Numbers: $-7$ and $+2$ (since $-7 \times 2 = -14$ and $-7 + 2 = -5$)
$x^2 - 5x - 14 = (x - 7)(x + 2)$
Answer: $(x - 7)(x + 2)$
$a = 2$, $b = 5$, $c = -3$
$b^2 - 4ac = 25 - 4(2)(-3) = 25 + 24 = 49$
$x = \frac{-5 \pm \sqrt{49}}{2(2)} = \frac{-5 \pm 7}{4}$
$x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$
$x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$
Answer: $x = \frac{1}{2}$ or $x = -3$
$2x + y = 7$
$x - y = 2$
$(2x + y) + (x - y) = 7 + 2$
$3x = 9$
$x = 3$
$3 - y = 2$
$y = 1$
Answer: $(3, 1)$
$x^2 - 9 = (x+3)(x-3)$
$x^2 - x - 6 = (x-3)(x+2)$
$\frac{(x+3)(x-3)}{(x-3)(x+2)} = \frac{x+3}{x+2}$
Answer: $\frac{x+3}{x+2}$ where $x \neq 3, -2$
💡 Tips & Tricks
Plug your solution back into the original equation. This catches sign errors and arithmetic mistakes—the most common errors in algebra.
Don't skip steps in your head. Writing each step helps prevent errors and makes it easier to find mistakes if your answer is wrong.
Remember the order of operations: Parentheses, Exponents, Multiplication & Division (left to right), Addition & Subtraction (left to right).
For $x^2 + bx + c$, look for two numbers that multiply to c and add to b. This makes factoring much faster!
Factoring: When numbers are nice and factors are obvious
Quadratic Formula: When factoring seems difficult or for decimals
Completing the Square: When you need vertex form or exact values
Before solving a quadratic, check $b^2 - 4ac$:
• Positive → 2 real solutions
• Zero → 1 real solution (double root)
• Negative → No real solutions (complex numbers)
⚠️ Common Mistakes to Avoid
Wrong: $-(3x - 5) = -3x - 5$
Right: $-(3x - 5) = -3x + 5$
Remember: The negative sign distributes to EVERY term inside!
Wrong: $-2x > 6 \Rightarrow x > -3$
Right: $-2x > 6 \Rightarrow x < -3$
When dividing or multiplying by a negative, flip the inequality!
Wrong: $(x + y)^2 = x^2 + y^2$
Right: $(x + y)^2 = x^2 + 2xy + y^2$
You cannot distribute exponents over addition!
Problem: $\frac{x^2 - 4}{x - 2} = x + 2$
Remember: This is only valid when $x \neq 2$
Always check for values that make the denominator zero!
Wrong: $\frac{x + 3}{3} = x$
Right: $\frac{x + 3}{3} = \frac{x}{3} + 1$
You can only cancel factors, not terms that are added!
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