Equation x^p - 1 = 0 {cyclotomic equation}, where p is prime number and x is independent variable, constructs polygons in circles. Polygons in circles have rotation-symmetry groups.
Lower-order equations {defective equation} can derive from equations.
Polynomial equations {Diophantine equation} can have integer coefficients. It can have integer solutions. If system of Diophantine equations has nine variables, no algorithm can decide if system has integer solutions.
Linear equations {homogeneous equation}| can have constant equal zero.
Equations {parametric equation} can depend on one variable. For example, line has parametric equations x = x1 + t*A and y = y1 + t*B, where t is parameter, A and B are constants, x and y are coordinates, and (x1,y1) is point on line. Line slope is B/A, if A is not 0. If A = 0, line is parallel to y-axis.
Wave equation {periodic equation} repeats at regular intervals: y = A * sin(a*x + b) or y = A * cos(a*x + b) or y = A * sin(a*x + b) + B * cos(c*x + d), where A is maximum y amplitude, a and b are constants, and x is independent variable. Phase angle is a*x + b. Angle phase shift from 0 is -b/a.
Two ratios can be equal {proportion} {analogy, mathematics}: a/b = c/d. In proportion, variable can be in numerator and another variable can be in other numerator {direct variation} {direct proportion}: x/3 = y/4. In proportions, product of two variables can be in numerator {inverse variation} {inverse proportion}: x*y = 4. Direct or inverse proportion can have a constant {proportionality constant}: c * (x/3) = y/4 or c * (x*y) = 4.
Proportions can change to product equalities by multiplying {cross-multiply} both equation sides by denominator product: x/3 = y/4 goes to 12 * x/3 = 12 * y/4 = 4*x = 3*y.
Two-variable equations {symmetric equation} can interchange x and y to make same equation.
Equations {quadratic equation} {second-degree equation} {quadric equation} can have one variable raised to second power. Quadratic equations have form a*x^2 + b*x + c = 0. Solution is x = (-b + (b^2 - 4*a*c)^0.5)/(2*a) and x = (-b - (b^2 - 4*a*c)^0.5)/(2*a). Quantity under square root is b^2 - 4*a*c {discriminant, quadratic equation}.
square
If quadratic equation has form a^2 * x^2 + b * x = c, add (b/(2*a))^2 to both equation sides {completing the square}. Factor left side: (a*x + b/(2*a))^2 = c. Take square root of both sides: a*x + b/(2*a) = c^0.5. Solve for x: x = (-b/(2*a) + c^0.5)/a.
factoring
If quadratic equation has form a*(x - r1)*(x - r2) = 0, then x = r1, x = r2, a = r1 + r2, and r1 * r2 = c/a.
graph
Quadratic-equation graph has U-shape.
Equations {cubic equation} can have variable raised to third power. General cubic equation has a solution.
Equations {quartic equation} can have variable raised to fourth power. General quartic equation has a solution.
General degree-higher-than-four equations {n-tic equation} do not have solutions.
Equations {linear equation} {first-degree equation} can have variables raised only to first power.
graph
Linear-equation graph is a straight line.
form
Standard form is a1 * x1 + a2 * x2 + ... + aN * xN = b, where b is constant, a1 to aN are coefficients, and x1 to xN are variables.
homogeneous
Linear homogeneous equations equal zero.
one variable
Linear equation can have only one variable: a1 * x1 = b. Straight line can have point where line intercepts y-axis {y-intercept} and point where line intercepts x-axis {x-intercept}. Straight line inclines to x-axis {slope, line}.
Linear equation can have form a*y + b*x = c {general form}. x = (c - a*y) / b. Slope is -b/a. y-intercept is c/a.
Linear equation can have form x/a + y/b = 1 {intercept form, line}. y-intercept is a. x-intercept is b.
Straight lines in planes can have length p. x * cos(A) + y * sin(A) - p = 0 {normal equation}, where A is angle that perpendicular from origin (0,0) to straight line makes with positive x-axis.
Linear equation can have form y - A = m * (x - B) {point-slope form}. (B,A) is line point. Slope is m.
Linear equation can have form y = m*x + b {slope-intercept form}. x = (y - b) / m. Slope is m. y-intercept is b.
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Date Modified: 2022.0225