Algebra
Free lessons and practice worksheets for algebra — worked examples, common mistakes, and step-by-step solutions.
- 01Algebra3 min read
Exponential Growth & Decay
Exponential growth and decay describe situations where quantities multiply by a constant factor over equal time periods. The general form y = a · bˣ captures this pattern, where a represents the initial amount, b is the growth factor, and x is the number of time periods. When b > 1, the quantity grows exponentially; when 0 < b < 1, it decays exponentially.
- 02Algebra3 min read
Exponents & Powers
An exponent represents repeated multiplication, where a base number is multiplied by itself a specified number of times. The expression 2³ means 2 × 2 × 2, which equals 8. Exponents follow specific rules that make calculations with large numbers more manageable, such as 2⁴ × 2³ = 2⁷ = 128.
- 03Algebra3 min read
Inequalities
An inequality compares two expressions using symbols like <, >, ≤, or ≥ instead of an equals sign. Solving inequalities follows the same steps as solving equations, with one crucial exception: multiplying or dividing both sides by a negative number flips the inequality sign. For example, -2x > 6 becomes x < -3 after dividing by -2.
- 04Algebra3 min read
Linear Equations
A linear equation contains a variable raised to the first power and forms a straight line when graphed. The goal is to isolate the variable by performing the same operation on both sides of the equation. Linear equations appear in forms like x + 5 = 12 or 3x - 7 = 14, where the variable has no exponents or radicals.
- 05Algebra3 min read
Logarithms
A logarithm is the inverse operation of exponentiation, answering the question of what power a base must be raised to in order to produce a given result. The notation log_b(x) = n means that b^n = x, where b is the base, x is the argument, and n is the result. For example, log_2(8) = 3 because 2^3 = 8.
- 06Algebra3 min read
Polynomials
A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents combined using addition and subtraction. The expression 3x² + 2x - 7 represents a polynomial with degree 2, where the highest power of the variable determines the degree. Polynomials appear throughout algebra and serve as building blocks for more advanced mathematical concepts covered in CCSS.HSA.APR standards.
- 07Algebra3 min read
Quadratic Equations
A quadratic equation is a polynomial equation of degree 2, written in standard form as ax² + bx + c = 0, where a ≠ 0. These equations have at most 2 solutions, which represent the x-intercepts of a parabola when graphed. The solutions can be found through factoring, completing the square, or the quadratic formula.
- 08Algebra3 min read
Scientific Notation
Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10, written in the form c × 10^n. The number 450,000 becomes 4.5 × 10^5, while 0.0032 becomes 3.2 × 10^-3. This system standardizes number representation across all magnitudes, from atomic scales to astronomical distances.
- 09Algebra3 min read
Systems of Equations
A system of equations consists of two or more equations that share the same variables and must be satisfied simultaneously. The solution represents the point where all equations intersect, typically expressed as an ordered pair (x, y) for two-variable systems. Systems appear throughout algebra, with CCSS 8.EE introducing basic solving methods and CCSS HSA.REI extending to more complex applications.
- 010Algebra3 min read
Two-Step Equations
A two-step equation contains one variable term and one constant term, requiring exactly two inverse operations to solve. These equations follow the pattern ax + b = c or ax - b = c, where the coefficient a and constant b can be any real numbers. The solving process systematically undoes operations in reverse order to isolate the variable.