Equation Solver
Solve linear, quadratic, cubic, and polynomial equations with step-by-step solutions. Enter values for instant results with step-by-step formulas.
Calculator
Adjust values & calculateStep-by-Step Solution
Formula
The quadratic formula solves ax^2 + bx + c = 0. For linear equations ax + b = 0, x = -b/a. For cubic equations, the Rational Root Theorem and synthetic division reduce the problem to a quadratic. The discriminant (b^2 - 4ac) determines whether roots are real or complex.
Last reviewed: December 2025
Worked Examples
Example 1: Solve a Quadratic Equation
Example 2: Solve a Cubic Equation
Background & Theory
The Equation Solver applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Equation Solver traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Key Features
- Solves linear, quadratic, and higher-degree polynomial equations step by step, returning all real and complex roots with full working shown.
- Simplifies fractions to lowest terms and computes ratios and proportions, including cross-multiplication checks and equivalent fraction generation.
- Performs complete prime factorization of any integer and computes the Greatest Common Divisor and Least Common Multiple for sets of numbers.
- Handles matrix operations including addition, scalar multiplication, matrix multiplication, determinant calculation, and full matrix inversion for square matrices.
- Evaluates all standard trigonometric functions and their inverses in degrees or radians, and verifies common trigonometric identities symbolically.
- Calculates permutations, combinations, and binomial coefficients for combinatorics problems, supporting both formula display and step-by-step breakdown.
- Converts integers between binary, octal, decimal, and hexadecimal bases instantly, with optional display of the positional value expansion.
- Computes the sum of arithmetic and geometric sequences given the first term, common difference or ratio, and number of terms, with formula derivation.
Frequently Asked Questions
Formula
Quadratic: x = (-b +/- sqrt(b^2 - 4ac)) / 2a
The quadratic formula solves ax^2 + bx + c = 0. For linear equations ax + b = 0, x = -b/a. For cubic equations, the Rational Root Theorem and synthetic division reduce the problem to a quadratic. The discriminant (b^2 - 4ac) determines whether roots are real or complex.
Worked Examples
Example 1: Solve a Quadratic Equation
Problem: Solve x^2 - 5x + 6 = 0
Solution: Using the quadratic formula: a=1, b=-5, c=6\nDiscriminant = (-5)^2 - 4(1)(6) = 25 - 24 = 1\nSince discriminant > 0, two real roots exist\nx = (5 + sqrt(1)) / 2 = 6/2 = 3\nx = (5 - sqrt(1)) / 2 = 4/2 = 2\nVerification: 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0\nVerification: 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0
Result: x = 3 and x = 2 (sum of roots = 5, product = 6)
Example 2: Solve a Cubic Equation
Problem: Solve x^3 - 6x^2 + 11x - 6 = 0
Solution: Try rational roots (factors of 6): x=1, x=2, x=3\nTest x=1: 1 - 6 + 11 - 6 = 0 (root found!)\nSynthetic division by (x-1): x^2 - 5x + 6 = 0\nSolve quadratic: discriminant = 25 - 24 = 1\nx = (5+1)/2 = 3 and x = (5-1)/2 = 2\nAll three roots: x = 1, x = 2, x = 3
Result: x = 1, x = 2, and x = 3
Frequently Asked Questions
What is a linear equation and how do I solve it?
A linear equation is an equation where the variable appears only to the first power, taking the general form ax + b = 0 where a and b are constants. To solve a linear equation, isolate the variable x by performing inverse operations on both sides. Subtract b from both sides to get ax = -b, then divide both sides by a to get x = -b/a. Linear equations always have exactly one solution unless a equals zero, in which case the equation is either an identity (true for all x when b also equals zero) or a contradiction (no solution when b is not zero). Linear equations represent straight lines when graphed and are the foundation for more complex equation types.
What does the discriminant tell us about an equation?
The discriminant is the expression under the square root in the quadratic formula, calculated as b squared minus 4ac. It reveals crucial information about the nature and number of solutions without actually solving the equation. A positive discriminant means the equation has two distinct real roots and the parabola crosses the x-axis at two points. A discriminant of zero means there is exactly one real root (a repeated root) and the parabola touches the x-axis at its vertex. A negative discriminant means there are no real roots, only two complex conjugate roots, and the parabola does not intersect the x-axis at all. The discriminant is also useful for analyzing conic sections and higher-degree polynomials.
How do I solve a cubic equation?
Solving cubic equations of the form ax cubed plus bx squared plus cx plus d equals zero typically involves finding one rational root using the Rational Root Theorem, then reducing to a quadratic equation through synthetic division or polynomial long division. The Rational Root Theorem states that any rational root p over q must have p as a factor of the constant term d and q as a factor of the leading coefficient a. Once you find one root, divide the cubic by the factor (x minus that root) to get a quadratic, which you solve using the quadratic formula. If no rational roots exist, numerical methods like Newton-Raphson iteration or Cardano formula for the depressed cubic can find the roots.
What is the difference between an equation and an expression?
An equation contains an equals sign and states that two expressions are equal, such as 2x plus 3 equals 7, and it can be solved to find specific values of the variable that make the statement true. An expression is a mathematical phrase without an equals sign, such as 2x plus 3, and it cannot be solved but can be simplified or evaluated for specific values of x. Equations have solutions (specific values), while expressions have values that depend on the variable. You can simplify expressions by combining like terms and applying operations, but solving requires an equation with two sides. Equation Solver works with equations, finding the values of x that make the left side equal to zero.
How do I verify Equation Solver's result independently?
The Formula section on this page shows the equation used. You can reproduce the calculation manually or in a spreadsheet using those steps. Compare your answer against the worked examples in the Examples section, which use known reference values so you can confirm the calculator is behaving as expected.
Can I use Equation Solver on a mobile device?
Yes. All calculators on NovaCalculator are fully responsive and work on smartphones, tablets, and desktops. The layout adapts automatically to your screen size.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy