CalculateMath

Complex Number Addition Calculator Add Two Complex Numbers Enter two complex numbers in the form a + bi and click Calculate . Example: 3 + 2i, 1 - 5i Calculate Sum Result will appear here. Note: Adding complex numbers means adding the real parts together and the imaginary parts together. Complex Number Addition – Complete Explanation A complex number is a number of the form a + bi , where a is the real part , b is the imaginary part , and i is the imaginary unit with the property i² = -1 . Adding complex numbers is straightforward: simply add the corresponding real parts and imaginary parts separately. Complex Addition Rule (a + bi) + (c + di) = (a + c) + (b + d)i Addition is commutative and associative: z₁ + z₂ = z₂ + z₁ and (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃) 1. Steps to Add Complex Numbers Identify the real parts of both complex numbers Identify the imaginary parts of both complex numbers Add the real parts together Add the ...

Partial Fraction Decomposition Calculator Decompose a Rational Function into Partial Fractions Enter a rational function P(x)/Q(x) (numerator degree < denominator degree) and click Decompose . The calculator provides step-by-step partial fraction decomposition. Decompose Decomposition result will appear here. Note: This calculator handles simple linear factors. For repeated or quadratic factors, follow the examples below. Partial Fraction Decomposition – Complete Explanation Partial fraction decomposition expresses a rational function as a sum of simpler fractions, making integration and algebraic manipulation easier. This method is essential in calculus, differential equations, and engineering. Partial Fraction Principle For a proper rational function P(x)/Q(x), if Q(x) factors into linear factors, we can write: P(x)/Q(x) = A/(factor1) + B/(factor2) + ... Constants A, B, etc., are determined by solving equations. 1. Steps for Decom...

Polynomial Remainder Calculator Find the Remainder of a Polynomial Division Enter a polynomial P(x) and a divisor x − a (example: P(x) = x³ − 6x² + 11x − 6, x − 2 ) and click Calculate Remainder . Calculate Remainder Remainder will appear here. Note: The remainder is the value left after dividing the polynomial by the divisor. You can also use the Remainder Theorem for divisors of the form x − a . Polynomial Remainder – Complete Explanation The Remainder Theorem is one of the most useful shortcuts in algebra. It states that if a polynomial P(x) is divided by x − a, then the remainder is P(a). This provides a fast way to find remainders without performing long division. The Remainder Theorem If a polynomial P(x) is divided by (x − a), the remainder is P(a). If P(a) = 0, then (x − a) is a factor of P(x). 1. Steps to Use the Remainder Theorem Write the polynomial P(x) in standard form (descending powers) Identify 'a...

Quadratic Inequality Solver Solve Quadratic Inequalities Enter a quadratic inequality (example: x² − 5x + 6 > 0 ) and click Solve to understand the solution process. 0" value="x^2 - 5x + 6 > 0"> Solve Solution will appear here. Note: Quadratic inequalities are solved by finding critical points and testing intervals, not by finding a single value. Quadratic Inequalities – Complete Explanation A quadratic inequality is an inequality that involves a quadratic expression, usually written in the form: ax² + bx + c > 0 ax² + bx + c ≥ 0 ax² + bx + c < 0 ax² + bx + c ≤ 0 Instead of finding one solution, we determine the range of values for which the quadratic expression is positive, negative, or zero. This comprehensive guide will take you through the complete process of solving quadratic inequalities, from basic concepts to advanced applications. 1. Key Idea Behind Quadratic Inequalities Quadratic expressions f...

Solve System of Inequalities System of Inequalities Solver Enter multiple inequalities (one per line), for example: x > 2 x ≤ 5 x > 2 x ≤ 5 Solve System Solution will appear here. Note: The solution of a system of inequalities is the set of values that satisfy all inequalities at the same time (the intersection of all intervals). System of Inequalities – Complete Explanation A system of inequalities is a group of two or more inequalities that are solved together. The solution is the set of all values that satisfy every inequality in the system. This comprehensive guide will take you through everything you need to know about solving systems of inequalities, from basic concepts to advanced applications. 1. What Is a System of Inequalities? Just like systems of equations, systems of inequalities involve multiple conditions. However, instead of a single value, the solution is usually a range of values (or region) that satisfies all c...

Solve Inequalities with One Variable One-Variable Inequality Solver Enter a linear inequality (example: 2x − 5 > 7 ) and click Solve to see the solution steps. 7" value="2x - 5 > 7"> Solve Solution will appear here. Note: This tool demonstrates the solving process. Always pay attention to inequality signs, especially when multiplying or dividing by negatives. Solve Inequalities – Complete Explanation An inequality is a mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥. Solving inequalities with one variable means finding all values of the variable that make the inequality true. This comprehensive guide will take you from basic inequality concepts through advanced problem-solving techniques, with plenty of examples and practice opportunities. 1. What Is an Inequality? Unlike equations, inequalities do not have a single solution. Instead, they usually have a range of soluti...

Expand Binomials Calculator Binomial Expansion Tool Enter two binomials (for example: (x + 2)(x + 3) ) and click Expand to see the full step-by-step expansion. Expand Expanded result will appear here. Note: This calculator focuses on demonstrating the expansion process. Use clear parentheses and standard algebraic notation for best results. Expand Binomials – Complete Mathematical Explanation Expanding binomials is one of the most important skills in algebra. A binomial is an expression with exactly two terms, such as x + 3 or 2x - 5 . When we expand binomials, we rewrite a product of binomials as a single polynomial. This skill is essential for simplifying expressions, solving equations, and preparing for advanced topics such as quadratic functions, calculus, and algebraic modeling. 1. What Does It Mean to Expand? To expand an algebraic expression means to remove parentheses by multiplying each term inside one set of parentheses by each ...