Algebra I

A quadratic equation is a polynomial equation of degree 2 — the highest power of the variable is 2. Its graph is a parabola, and solving it means finding where that parabola crosses the x-axis. The quadratic formula gives a universal method for solving any quadratic equation, no matter how ugly its coefficients.

An absolute value equation contains a variable expression inside absolute value bars, such as |x - 3| = 7. Because |a| represents distance from zero, an absolute value equation typically splits into two separate linear equations — one for the case where the inside expression is positive, and one where it's negative — each of which must be solved separately.

An inequality compares two expressions using <, >, ≤, or ≥ instead of an equals sign, and its solution is typically a whole RANGE of values rather than a single number. Solving a linear inequality uses the same balance-scale operations as solving an equation, with one crucial extra rule: multiplying or dividing both sides by a negative number reverses the inequality's direction.

A linear equation is an equation in which every variable term appears only to the first power and is never multiplied by another variable — its graph is always a straight line (in two variables) or a single point/line/plane in general. Linear equations are the simplest and most widely used equations in mathematics: they model constant rates of change, describe straight-line relationships between quantities, and form the building blocks for nearly every more advanced algebraic structure, from systems of equations to linear algebra itself.

A linear inequality in two variables, such as y < 2x + 3, has a solution set that is an entire region of the coordinate plane rather than a single line. The boundary line (from the corresponding equation) divides the plane into two half-planes, and the inequality's solution is one of those half-planes, either including or excluding the boundary line itself.

A system of inequalities is a set of two or more inequalities considered together. Its solution is the set of all points satisfying EVERY inequality simultaneously — graphically, the overlapping region where all the individual shaded half-planes intersect. Systems of inequalities are the foundation of linear programming, where a feasible region defined by constraints is searched for an optimal value.

A system of linear equations is a set of two or more linear equations considered together, sharing the same variables. Solving the system means finding the value(s) of every variable that satisfy ALL the equations simultaneously. Geometrically, in two variables, each equation is a line, and a solution is a point where the lines intersect. Systems of equations model any situation where multiple constraints must hold at once — mixing problems, break-even points, and network flows.

Completing the square is a technique for rewriting a quadratic expression ax² + bx + c as a perfect square trinomial plus (or minus) a constant, a(x - h)² + k. This single algebraic maneuver unlocks the vertex of a parabola directly, gives the cleanest possible derivation of the quadratic formula, and is the essential first step for converting circle, ellipse, and hyperbola equations into standard form. It converts a quadratic that's awkward to reason about into one whose maximum/minimum value and axis of symmetry are immediately visible.

The quadratic formula gives the solutions to ANY quadratic equation ax² + bx + c = 0 directly in terms of its coefficients a, b, and c — no factoring, guessing, or graphing required. It is one of the most celebrated results in elementary algebra precisely because it is universal: unlike factoring, which only works cleanly for equations with 'nice' rational roots, the quadratic formula solves every quadratic equation, including those with irrational or complex solutions, by a single fixed procedure.

Two quantities vary directly if their ratio is constant (one grows proportionally as the other grows), and vary inversely if their product is constant (one shrinks proportionally as the other grows). These two relationships describe an enormous number of real-world quantities — from speed and travel time to pressure and volume — using just a single constant of proportionality.

The domain of a function is the complete set of input values (typically x-values) for which the function is defined; the range is the complete set of output values (y-values) the function actually produces. Together they describe exactly what a function 'accepts' and 'returns' — essential for knowing where a function's graph exists and what values it can take.

Function transformations describe how the graph of a function changes when you modify its equation — shifting it up, down, left, or right, stretching or compressing it, or reflecting it across an axis. Learning these transformation rules lets you sketch complicated-looking functions instantly by starting from a simple parent function (like y=x² or y=√x) and applying a short sequence of predictable moves.

Factoring is the process of rewriting a polynomial as a product of simpler polynomials — the reverse of expanding/multiplying. It is one of the most useful skills in algebra because it converts sums (which are hard to solve or simplify) into products (whose zeros and common factors are easy to read off). Factoring is essential for solving polynomial equations, simplifying rational expressions, and revealing the structure hidden inside an expression.

A polynomial is an expression built from variables and constants using only addition, subtraction, and non-negative integer exponents — no division by a variable, no variables under a radical, and no negative or fractional exponents on variables. Polynomials are the workhorse expressions of algebra: every quadratic, cubic, and higher-degree equation studied in school algebra is a polynomial equation, and polynomial functions are the smoothest, best-behaved functions in mathematics.

A radical expression is an expression containing a root — a square root, cube root, or higher-order root — of a variable or number. Radicals are the inverse operation of exponentiation: √x 'undoes' x², and more generally the n-th root undoes the n-th power. Simplifying and combining radical expressions correctly is essential for working with irrational numbers, the Pythagorean theorem, and the quadratic formula.

A rational expression is a ratio of two polynomials, such as (x+3)/(x²-9), analogous to how a rational NUMBER is a ratio of two integers. Rational expressions appear whenever a quantity is described as one polynomial divided by another — rates, densities, and averages. Because the denominator is a polynomial, values that make it zero must be excluded from the domain, and simplifying a rational expression almost always begins with factoring both numerator and denominator.

Graphing a linear equation means drawing the straight line that represents all of its solutions in the coordinate plane. Because a line is entirely determined by two points (or one point and a slope), graphing a linear equation reduces to finding just enough information — an intercept, a slope, or two solution pairs — to draw the line accurately.

Slope is a number that measures the steepness and direction of a line: how much the y-value changes for every unit increase in x. It is the discrete, algebraic ancestor of the derivative — a constant rate of change rather than an instantaneous one — and it is the single most important number describing a linear relationship between two variables. Slope tells you whether a line rises or falls, how steeply, and lets you write the equation of any line given just a point and a direction.

Slope-intercept form writes the equation of a line as y = mx + b, where m is the slope and b is the y-intercept. It is the most convenient form for graphing a line quickly, since both the starting point (the y-intercept) and the direction (the slope) can be read off directly from the equation without any computation.