Equation solutions {root, equation}| {zero, equation} are variable values that make equation true {solution, equation} {equation, solving}. Roots {null, equation} can be non-existent or zero. Equation-solution number equals equation degree. Two or more solutions can be equal {multiplicity, solution}.
order
Equations have highest exponent. Roots can be highest radical {order, radical} {radical, order}.
homogeneity
Equations can equal zero: f(x) = 0. Homogeneous equations have solutions.
polynomial
If polynomials have real-number coefficients, one factor is coefficient of highest-power term times product of polynomials of form (x - r) or (x^2 - (r + r')*x + r*(r')), where r is root and r' is root complex conjugate. Root complex conjugate is also a root.
complex number
If x and n are real, equation x^n = -1 solutions are complex numbers {roots of unity} {nth roots of unity}. For x^2 = -1, x = i. For x^3 = -1, x = i^(2/3). If w and z are complex, equation w^z = 1 solutions are log(1) = 0, 2 * pi * i, 4 * pi * i, 6 * pi * i, 8 * pi * i, ...
In unit circle, solutions are regular-polygon vertices, forming complex-number cyclic group Zn, a finite multiplicative group.
If equations {determinate equation} have only one unknown variable, equation has a numerical solution. If equation {indeterminate equation} has more than one unknown variable, equation has solution in terms of unknowns. If unknowns number equals equation number, equation system has numerical solutions. If unknowns number is more than equation number, equation system has solutions in terms of unknowns.
Expressions can have numeric values {evaluation, expression}.
To check solution steps {checking solution}, do additions and multiplications forward and backward, making sure that sign is correct. For (x - 3) / (x + 2) = x / (x - 2), x^2 - 2*x - 3*x + 6 = x^2 + 2, so 7x = 6.
For general quadratic-equation solution, quantity {discriminant, solution} under square root sign is b^2 - 4*a*c.
If unknowns number is less than equation number, solutions {extraneous solution}| can depend on other solutions.
For polynomials divisible by (x - a)^n, root a {repeated root} repeats n times.
If polynomial equals zero and has two positive or negative terms in succession, at least one root is negative {signs and roots rule} {rule of signs and roots}. If polynomial equals zero and has positive term succeeding negative term, or negative term succeeding positive term, at least one root is positive.
If you know two function points, two points make a line, and you can use the linear function to estimate function values {extrapolation}| for independent-variable values greater than the larger, or less than the smaller, of the point independent-variable values.
If you know two function points, two points make a line, and you can use the linear function to estimate function values {interpolation}| for independent-variable values between the point independent-variable values.
All terms with unknown variable can be on left or right equation side {isolate variable}.
process
Remove exponents outside parentheses by multiplying.
Remove all fractions and divisions by multiplying out all denominators. Factor denominator and cancel factors, divide into numerator to get quotient, or multiply both equation sides by denominator.
Remove parentheses by performing all multiplications, to make sum of terms.
Make irrational numbers or variables rational by taking both equation sides to power.
Multiply repeated variables in terms to make one variable.
Add similar terms.
Add all constants.
Put all terms containing unknown variable on one equation side.
roots
Assume left-side expression has unknown variable. Find left-side-expression roots by polynomial factoring or other method. Find right-side-expression roots. For whole equation, root is left-side-expression root minus right-side-expression root.
If h is small compared to x, change (x + h)^n to x + n*h to remove exponent and make linear {linearize}.
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Date Modified: 2022.0225