Equation Solver
Solve linear and quadratic equations step by step. Enter coefficients and see the solution with explanation.
Key features
- Linear equations (ax+b=0)
- Quadratic (ax^2+bx+c=0)
- Step-by-step solution
- Complex roots support
Guide
Solving mathematical equations means finding the values of unknown variables that make the equation true. The process ranges from simple one-step arithmetic to multi-step algebraic manipulation. This guide covers the types of equations you encounter in academic and practical settings, the solving methods for each type, common pitfalls, and how to verify your solutions. A linear equation in one variable has the form ax + b = c, where x is the unknown and a, b, c are known numbers. Solving requires isolating x on one side of the equation using inverse operations. For 3x + 7 = 22, subtract 7 from both sides to get 3x = 15, then divide both sides by 3 to get x = 5. Linear equations always have exactly one solution (unless the coefficients create a contradiction like 0x = 5, which has no solution, or an identity like 0x = 0, which has infinitely many solutions). The fundamental principle of equation solving is that you can perform any operation on both sides of an equation without changing the solution, as long as the operation is valid. You can add, subtract, multiply, or divide both sides by the same number (except dividing by zero). You can apply the same function to both sides (square root, logarithm, etc.) with appropriate care about domain restrictions. Every step maintains the equality while moving closer to isolating the unknown variable. Multi-step linear equations require several operations in sequence. For 2(3x - 4) + 5 = 23, first distribute: 6x - 8 + 5 = 23. Combine like terms: 6x - 3 = 23. Add 3 to both sides: 6x = 26. Divide by 6: x = 26/6 = 13/3. Each step simplifies the equation while preserving the solution. The order of operations in reverse (undo addition/subtraction first, then multiplication/division, then exponents) guides which operation to apply at each step. Equations with variables on both sides require collecting all variable terms on one side and all constant terms on the other. For 5x + 3 = 2x + 18, subtract 2x from both sides: 3x + 3 = 18. Subtract 3: 3x = 15. Divide by 3: x = 5. It does not matter which side you collect variables on. Choose whichever keeps the coefficient positive to reduce sign errors. Quadratic equations have the form ax^2 + bx + c = 0. They can have two solutions, one solution, or no real solutions. Three main solving methods exist. Factoring works when the quadratic factors neatly: x^2 - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0, giving x = 2 and x = 3. The quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / (2a) works for any quadratic. Completing the square rewrites the equation in vertex form and is useful for understanding the geometry of the parabola. The discriminant (b^2 - 4ac) determines the nature of quadratic solutions. If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution (a repeated root). If it is negative, there are no real solutions (the solutions are complex numbers involving the imaginary unit i). Checking the discriminant before solving tells you what to expect and helps catch errors. If you are getting two solutions but the discriminant is zero, something went wrong. Systems of linear equations involve two or more equations with two or more unknowns. A system of two equations in two variables (x and y) can be solved by substitution (solve one equation for one variable, substitute into the other), elimination (add or subtract equations to eliminate one variable), or graphing (find the intersection point). For the system x + y = 10 and 2x - y = 5, adding the equations gives 3x = 15, so x = 5, and substituting back gives y = 5. Equations with fractions are solved by finding the least common denominator (LCD) of all fractions in the equation and multiplying every term by the LCD. This eliminates all fractions and produces a simpler equation. For x/3 + x/4 = 7, the LCD is 12. Multiplying through: 4x + 3x = 84, so 7x = 84 and x = 12. Always check your answer in the original equation because multiplying by expressions containing the variable can introduce extraneous solutions. Equations with absolute values require considering two cases because |a| = b means a = b or a = -b. For |2x - 3| = 7, case 1: 2x - 3 = 7, giving x = 5. Case 2: 2x - 3 = -7, giving x = -2. Both solutions should be verified. If the equation is |2x - 3| = -7, there is no solution because absolute values are never negative. Equations with absolute values on both sides, like |x + 1| = |2x - 3|, generate four cases but usually only two are distinct. Radical equations contain variables under a square root or other root. To solve sqrt(x + 3) = 5, square both sides: x + 3 = 25, so x = 22. Squaring both sides can introduce extraneous solutions, so always substitute back to verify. For sqrt(x + 3) = -5, there is no solution because a square root is never negative. For equations with multiple radicals, isolate one radical, square, then isolate and square again. Exponential equations have the variable in the exponent. For 2^x = 32, recognize that 32 = 2^5, so x = 5. For equations where both sides cannot be written with the same base, take logarithms of both sides. For 3^x = 20, take ln of both sides: x ln(3) = ln(20), so x = ln(20) / ln(3) = approximately 2.727. These equations model radioactive decay, population growth, compound interest, and signal attenuation. Logarithmic equations contain logarithms of expressions with the variable. For log(x + 5) = 2, convert to exponential form: x + 5 = 10^2 = 100, so x = 95. For equations with multiple logarithms, use log properties to combine them: log(a) + log(b) = log(ab), log(a) - log(b) = log(a/b), n log(a) = log(a^n). Always verify solutions because logarithms are undefined for non-positive arguments. If your solution makes any logarithm argument negative or zero, it is extraneous. Polynomial equations of degree 3 or higher (cubic, quartic, etc.) may have multiple real solutions. A cubic equation can have 1 or 3 real solutions. A quartic can have 0, 2, or 4 real solutions. For simpler cases, factoring works: x^3 - 6x^2 + 11x - 6 = 0 factors as (x-1)(x-2)(x-3) = 0, giving x = 1, 2, 3. For more complex polynomials, numerical methods like Newton's method find approximate solutions iteratively. Trigonometric equations like sin(x) = 0.5 have infinitely many solutions because trigonometric functions are periodic. The primary solutions are x = 30 degrees and x = 150 degrees (in the range 0 to 360 degrees). The general solutions are x = 30 + 360n and x = 150 + 360n degrees, where n is any integer. Solving trigonometric equations requires knowledge of the unit circle values and the periodicity of each function. Verifying solutions is a non-optional step. Substitute each solution back into the original equation and confirm both sides are equal. This catches arithmetic errors, extraneous solutions introduced by squaring or multiplying by variable expressions, and domain violations where a solution makes a denominator zero, a logarithm argument negative, or a square root argument negative. Professional mathematicians and engineers always verify. It takes 30 seconds and prevents costly errors. Word problems require translating English into mathematical equations before solving. The key is identifying the unknown (assign it a variable), identifying the relationships described in the problem (write them as equations), and solving the resulting equation(s). If the problem says a number plus 7 equals twice the number minus 3, that translates to x + 7 = 2x - 3, giving x = 10. Practice with word problems builds the ability to recognize mathematical structure in real-world situations. Common solving mistakes include sign errors when distributing negatives (distribute the negative to every term inside the parentheses, not just the first), forgetting to apply operations to both sides of the equation, dividing by a variable without considering the case where that variable equals zero, and algebraic errors in combining like terms. Working through each step carefully and writing down intermediate results helps prevent these errors. The WebRecast equation solver accepts equations in standard mathematical notation. Enter the equation with the variable you want to solve for, and the tool shows the step-by-step solution process. It handles linear equations, quadratic equations, equations with fractions, and polynomial equations. The step-by-step display serves as both a solving tool and a learning resource, showing the method so you can apply it independently. All computation happens in your browser with no external data transmission.
Frequently asked questions
Does it handle complex roots?
Yes, when the discriminant is negative, complex roots are shown.
