The art and science of solving equations is one of the oldest pursuits in mathematics, dating back to ancient Babylonian clay tablets from around 2000 BCE that contained solutions to what we now recognize as quadratic equations. The Babylonians used geometric methods, essentially completing the square through area manipulation, to find unknown quantities in problems about land measurement and commerce. Ancient Greek mathematicians, including Euclid and Diophantus, further developed equation-solving techniques, with Diophantus's "Arithmetica" from the 3rd century CE introducing symbolic abbreviations that moved algebra away from purely verbal descriptions.

The methods for solving linear equations are straightforward: isolate the variable by performing inverse operations on both sides of the equation, maintaining equality throughout. The process relies on the fundamental properties of equality and the field axioms of real numbers. For quadratic equations of the form ax squared plus bx plus c equals zero, several methods have been developed over millennia. Factoring works when the quadratic can be expressed as a product of two linear factors, but this is not always possible with rational coefficients. Completing the square, the method used by the Babylonians and later formalized by al-Khwarizmi in his 9th-century treatise "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (from which the word algebra derives), transforms any quadratic into a perfect square form that can be solved by taking square roots.

The quadratic formula, x equals negative b plus or minus the square root of b squared minus 4ac all divided by 2a, is derived directly from completing the square on the general quadratic equation. This formula provides solutions for any quadratic equation and introduces one of the most important concepts in algebra: the discriminant, defined as b squared minus 4ac. The discriminant determines the nature of the solutions without requiring their actual computation. When the discriminant is positive, the equation has two distinct real roots. When it equals zero, there is exactly one repeated real root (the parabola touches the x-axis at its vertex). When the discriminant is negative, there are no real solutions, but two complex conjugate roots exist in the complex number system, where the imaginary unit i represents the square root of negative one.

The substitution method for solving equations involves replacing one variable or expression with an equivalent form to simplify the equation. This technique extends beyond simple equations to systems of equations, where one equation is solved for a variable and substituted into another. Together, these algebraic methods form the foundation for more advanced equation-solving techniques used throughout mathematics, science, and engineering, from polynomial root-finding algorithms to numerical methods for equations that have no closed-form solutions.