Imaginary and complex numbers Imaginary numbers. Imaginary unit. Imaginary roots. Real numbers. Complex numbers. Consider the pure quadratic equation: x 2 = a , where a – a known value. Its solution may be presented as: Here the three cases are possible: 1). If a = 0 , then x = 0. 2). If a is a positive number, then its … [Read more...]

## Quadratic equation

Quadratic equation Quadratic equation. Reduced quadratic equation. Non-reduced quadratic equation. Pure quadratic equation. A quadratic equation is an algebraic equation of the second degree: ax 2 + bx + c = 0 , (1) where a, b, c – the given numerical or literal coefficients , x – an unknown. If a = 0, then this … [Read more...]

## Irrational numbers. Formula of complicated radical

Irrational numbers. Formula of complicated radical Rational numbers. Irrational numbers. Examples of irrational numbers. Formula of complicated radical. Irrational numbers in contrast to rational numbers (see above) aren’t presented as a vulgar, not cancelled fraction of the shape: m / n , where m and n are integers. There are numbers of a … [Read more...]

## Arithmetical root

Arithmetical root Arithmetical root. Algebraic root. Absolute value (modulus) of number. As we know, an even degree root has two values: positive and negative, so An arithmetical root of the n-th degree of a non-negative number a is called a non-negative number, the n-th power of which is equal to a . An algebraic root of the n-th degree of a given … [Read more...]

## Powers and roots

Powers and roots Operations with powers. Multiplication and division of powers. Power of product of some factors. Power of a quotient (fraction). Raising of power to a power. Operations with roots. Arithmetical root. Root of product of some factors. Root of quotient (fraction). Raising of root to a power. Proportional change of degrees of a root and its radicand. … [Read more...]

## Systems of three simultaneous linear equations in three unknowns

Systems of three simultaneous linear equations in three unknowns Systems of three simultaneous linear equations in three unknowns.Basic methods of solution. Substitution. Addition or subtractionof equations. The third order determinants. Cramer's rule. Systems of three simultaneous linear equations in three unknowns have the shape: … [Read more...]

## Systems of two simultaneous linear equations in two unknowns

Systems of two simultaneous linear equations in two unknowns Systems of two simultaneous linear equations in two unknowns. Basic methods of solution. Substitution. Addition or subtraction of equations. The second order determinants. Cramer's rule. Investigation of solutions. … [Read more...]

## Linear equations in one unknown

Linear equations in one unknown An equation of the shape: ax + b = 0, where a and b – the known numbers, x – an unknown value, is called a linear equation in one unknown. To solve this equation means to find the numerical value of x , at which … [Read more...]

## Main ways used at solving of equations

Main ways used at solving of equationsIdentical transformations. Replacement of expression. Transferring terms of equation from one side to another. Multiplication and division by non-zero expression (number). Raising to a power. Extraneous roots of equation. Extracting of a root. Loss of roots of equation. Solving of equation is a process, consisting mainly in a replacement … [Read more...]

## Equations: common information

Equations: common information Equality. Identity. Equation (unknowns, roots of an equation, solving). Equivalent equations. Equality - two expressions (numerical or literal ones), jointed by sign " = ". Identity - a valid numerical equality or a literal equality, valid at any numerical values of letters, contained in it. E x a m p l e s : 1) A numerical equality 4 · … [Read more...]