7 Equation Types | Step-by-Step Solutions | Real & Complex Roots
✓ 7 Equation Types - Linear to cubic
✓ Step-by-Step - Full working shown
✓ System Solver - 2 & 3 variables
✓ Complex Roots - Real & imaginary
10 Equation Types | Step-by-Step Solutions | Formula Explanations
📝 Form: ax + b = 0
First-degree polynomial with one variable
💡 Example: 2x + 3 = 0 → x = -1.5
Solve linear, quadratic, and cubic polynomial equations with detailed step-by-step solutions showing quadratic formula, discriminant analysis, and Cardano's formula for cubic.
Solve simultaneous equations with 2 or 3 variables using Cramer's rule and Gaussian elimination. Shows determinant calculations, matrix operations, and back substitution.
Solve exponential and logarithmic equations using natural logarithm properties, exponential laws, and algebraic manipulation with complete working steps.
An Equation Solver is an advanced mathematical tool designed to find solutions values of variables that satisfy given equations mathematical statements of equality between two expressions fundamental to algebra calculus engineering physics solving problems across mathematics sciences enabling students researchers engineers scientists to solve linear equations first-degree polynomials form ax plus b equals zero simplest type requiring single division operation yielding one solution or special cases infinite solutions when both coefficients zero no solution when only constant nonzero quadratic equations second-degree polynomials form ax squared plus bx plus c equals zero fundamental parabolic equations solved using quadratic formula x equals negative b plus minus square root of b squared minus four ac all over two a discriminant delta equals b squared minus four ac determining nature of roots delta greater than zero gives two distinct real roots parabola crosses x-axis twice delta equals zero gives one repeated root vertex touches x-axis delta less than zero gives complex conjugate roots parabola entirely above or below x-axis essential for projectile motion optimization problems quadratic relationships cubic equations third-degree polynomials form ax cubed plus bx squared plus cx plus d equals zero more complex requiring Cardano formula algebraic solution or Newton-Raphson numerical method iterative approximation converging to real root useful for volume calculations engineering design advanced physics system of linear equations multiple equations with multiple variables simultaneous satisfaction required two equations two unknowns x y solved using Cramer rule determinant method substitution elimination three equations three unknowns x y z requiring Gaussian elimination systematic row operations converting augmented matrix to row echelon form followed by back substitution finding all variable values critical for circuit analysis economic modeling chemical reactions equilibrium systems exponential equations containing exponential function e raised to power form a times e to bx equals c solved by taking natural logarithm both sides isolating variable important for growth decay problems population dynamics radioactive decay compound interest calculations logarithmic equations containing natural logarithm function form a times ln x plus b equals c solved by exponential transformation isolating logarithm then exponentiating pH calculations decibel measurements information theory applications making equation solver indispensable tool for mathematics education scientific research engineering practice.
Our Free Equation Solver 2025 offers comprehensive advanced features supporting seven equation types covering entire spectrum from basic linear equations to complex cubic polynomials simultaneous systems exponential logarithmic equations with linear equations ax plus b equals zero detecting special cases zero coefficient yielding infinite solutions identity equations or no solution contradictions providing clear explanations quadratic equations complete discriminant analysis showing nature of roots two distinct real one repeated real or complex conjugate pairs using standard quadratic formula accurate to six decimal places cubic equations combining Cardano theoretical formula with Newton-Raphson numerical iteration fifty iterations ensuring convergence to accurate approximation real root handling cases where algebraic solution impractical system of two equations using Cramer rule calculating main determinant determinant x determinant y checking for unique solution parallel lines no solution or dependent equations infinite solutions showing all intermediate determinant calculations making method transparent educational system of three equations employing Gaussian elimination with partial pivoting forward elimination creating upper triangular form followed by back substitution solving from bottom up handling singular matrices detecting inconsistent systems providing meaningful error messages exponential equations using natural logarithm properties log laws solving for variable in exponent checking validity of coefficients ensuring positive arguments logarithmic equations applying exponential function both sides using inverse relationship between ln and e verifying domain restrictions positive arguments unique features including step-by-step solution display showing every calculation intermediate step formula used reasoning behind each transformation perfect for learning understanding verifying homework exam preparation one-click example fill instantly populating fields with sample equations allowing quick testing experimentation without manual entry copy to clipboard functionality entire solution steps copied single click pasted into documents homework submissions reports mobile responsive design working flawlessly smartphones tablets desktops touch-friendly inputs adaptive layouts error handling comprehensive validation checking coefficient validity equation solvability providing clear helpful error messages guiding users correct input real and complex root display clearly showing real roots as decimals complex roots as a plus bi format discriminant values for quadratics making it most powerful user-friendly equation solver available online completely free unlimited calculations no registration advertisements serving high school students learning algebra college students tackling calculus engineering students solving technical problems researchers performing mathematical analysis educators teaching equation solving demonstrating methods anyone needing reliable accurate equation solving tool updated for 2025 academic standards latest mathematical algorithms pedagogical best practices.
For: ax² + bx + c = 0
Formula: x = (-b ± √(b² - 4ac)) / (2a)
Δ = b² - 4ac (discriminant)
Δ > 0: Two real roots | Δ = 0: One root | Δ < 0: Complex roots
For: a₁x + b₁y = c₁ and a₂x + b₂y = c₂
D = a₁b₂ - a₂b₁
x = (c₁b₂ - c₂b₁) / D
y = (a₁c₂ - a₂c₁) / D
D ≠ 0: Unique solution | D = 0: No/Infinite solutions
For: ax³ + bx² + cx + d = 0
Normalized to: x³ + px + q = 0
p = (3ac - b²) / (3a²)
q = (2b³ - 9abc + 27a²d) / (27a³)
Uses Newton-Raphson for numerical solution
For: a·e^(bx) = c
Divide by a:
e^(bx) = c/a
Take natural log:
bx = ln(c/a)
x = ln(c/a) / b
For: a·ln(x) + b = c
Subtract b:
a·ln(x) = c - b
Divide by a:
ln(x) = (c - b) / a
Exponentiate:
x = e^((c-b)/a)
Steps:
1. Form augmented matrix [A|b]
2. Forward elimination → Row Echelon Form
3. Back substitution from bottom
4. Solve for z, then y, then x
Pivot selection for numerical stability
Select equation type from dropdown, enter coefficients in input fields or click 'Fill Example' for sample values, click 'Solve' to get instant solution with step-by-step working. Solutions show real roots, complex roots, or special cases (infinite/no solution).
For ax² + bx + c = 0: Select 'Quadratic', enter a, b, c values, click Solve. Uses quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Shows discriminant (Δ = b² - 4ac): Δ > 0 → two real roots, Δ = 0 → one root, Δ < 0 → complex roots. Example: x² - 5x + 6 = 0 gives x = 2, 3.
Discriminant (Δ) = b² - 4ac determines root nature: Δ > 0: Two distinct real roots (parabola crosses x-axis twice), Δ = 0: One repeated root (vertex on x-axis), Δ < 0: Two complex conjugate roots (parabola doesn't cross x-axis). Calculator automatically calculates and explains.
For 2 equations (a₁x + b₁y = c₁, a₂x + b₂y = c₂): Select 'System of 2 Equations', enter all 6 coefficients, click Solve. Uses Cramer's Rule: D = a₁b₂ - a₂b₁, then x = Dₓ/D, y = Dᵧ/D. Shows unique solution, infinite solutions, or no solution. For 3 equations: uses Gaussian elimination.
Yes! For ax³ + bx² + cx + d = 0: Uses Cardano's formula combined with Newton-Raphson numerical method for accurate real root approximation. Cubic equations have 3 real or 1 real + 2 complex roots. Shows step-by-step with normalized form, discriminant, numerical iteration.
Complex roots occur when discriminant < 0. Format: a + bi (a = real part, b = imaginary part, i = √(-1)). Example: x² + x + 1 = 0 gives x = -0.5 ± 0.866i. Shows both conjugate pairs: x₁ = a + bi, x₂ = a - bi. Complex roots mean no real x-intercepts.
Shows complete working: 1) Original equation, 2) Formula/method used, 3) Value substitution, 4) Intermediate calculations, 5) Final answer. Each step numbered and explained. Perfect for learning, homework checking, exam prep. Can copy all steps to clipboard.
Yes! Exponential (a·e^(bx) = c): Uses natural log. Example: 2·e^(3x) = 10 → x ≈ 0.5365. Logarithmic (a·ln(x) + b = c): Uses exponential. Example: 2·ln(x) + 1 = 5 → x ≈ 7.389. Shows detailed steps with log laws, exponential properties.
Calculator detects: No Solution - parallel lines (D = 0 but inconsistent), contradiction like 0 = 5. Infinite Solutions - dependent equations (same line), identity like 0 = 0. Linear: 0x + 0 = 0 → infinite, 0x + 5 = 0 → no solution. Clear messages explain.
100% free forever! All 7 equation types completely free with unlimited calculations. No registration, no ads, no premium. Full step-by-step solutions, all methods (quadratic formula, Cramer's rule, Gaussian elimination, Newton-Raphson, Cardano), copy feature, example fills. Works on all devices!
Our Free Equation Solver 2025 is the most advanced comprehensive feature-rich online mathematical tool designed for students educators engineers scientists researchers offering unparalleled computational power solving seven distinct equation types from simple linear equations single variable first-degree polynomials to complex cubic equations third-degree polynomials simultaneous systems multiple variables exponential logarithmic equations transcendental functions with linear equations ax plus b equals zero simplest form detecting special cases zero coefficient infinite solutions contradictions no solution providing clear explanations educational value quadratic equations second-degree polynomials using standard quadratic formula discriminant analysis determining nature roots two distinct real one repeated real complex conjugate pairs showing complete calculation steps intermediate values perfect homework verification exam preparation understanding parabolic relationships cubic equations third-degree polynomials employing Cardano theoretical formula combined Newton-Raphson numerical iteration accurate approximation real roots handling complex algebraic solutions practical numerical methods system of two equations linear simultaneous two variables x y using Cramer rule determinant method calculating main determinant numerator determinants solving unique solutions detecting parallel lines inconsistent systems dependent equations infinite solutions system of three equations linear simultaneous three variables x y z employing Gaussian elimination partial pivoting forward elimination upper triangular form back substitution solving variables systematically handling singular matrices numerical stability exponential equations transcendental containing e to power form using natural logarithm properties solving variable exponent applications growth decay population dynamics radioactive decay compound interest logarithmic equations containing natural logarithm function using exponential inverse relationship domain restrictions pH calculations decibel measurements unique features step-by-step solution display every calculation intermediate step formula explanation reasoning educational transparency one-click example fill instant sample equations quick testing experimentation copy to clipboard functionality entire solution exportable single click pasting documents homework reports mobile responsive design flawless operation smartphones tablets desktops touch-friendly adaptive comprehensive error handling validation clear messages making it most powerful user-friendly equation solver online completely free unlimited calculations serving high school college engineering students researchers educators updated 2025 standards.