Equation Solvers for Algebra, Calculus, and Beyond

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a solves any equation of the form ax² + bx + c = 0. The discriminant b² - 4ac tells you the nature of the solutions before you compute them: positive = two real roots, zero = one repeated root, negative = two complex conjugate roots. Quadratic equations appear everywhere — projectile trajectories (height as a function of time), circuit analysis (resonance frequencies), economics (profit maximisation). The geometric interpretation: the roots are where the parabola y = ax² + bx + c crosses the x-axis. A negative discriminant means the parabola never touches the x-axis.

Linear equations have variables only to the first power: ax + b = c, or in two variables, ax + by = c (a line in the xy-plane). A system of linear equations is solved by substitution (express one variable in terms of others, substitute into another equation) or elimination (combine equations to cancel variables). For two equations in two unknowns, elimination is usually faster. For three or more, matrix methods (Cramer's rule, Gaussian elimination, matrix inverse) are systematic and scale to any size. Linear systems describe everything from circuit analysis (Kirchhoff's laws) to chemical balancing to traffic flow modelling.

Exponential equations have the variable in the exponent: 2^x = 32 or e^(kt) = N₀. To solve, take a logarithm of both sides: log(2^x) = log(32) → x · log(2) = log(32) → x = log(32)/log(2) = 5. Use natural log (ln) when the base is e, common log (log₁₀) for base 10, or change-of-base formula for other bases. Logarithmic equations like log(x) + log(x-3) = 1 are solved using log properties: log(a) + log(b) = log(ab), so log(x(x-3)) = 1 → x(x-3) = 10 → quadratic. Always verify the solution by substituting back — logarithms require strictly positive arguments, so some algebraic solutions may be extraneous.