Squaring both sides can introduce answers that fail
An equation joins two expressions with an equals sign invented by Robert Recorde in 1557. Identities hold for every allowed value; conditional equations hold only for some. Like a balance scale, whatever you do to one side you must do to the other—though some operations create extraneous solutions.
An equation is a formula stating two expressions are equal, connected by =. Robert Recorde invented the symbol in 1557, reasoning that parallel lines of equal length could represent nothing more equal. The left-hand and right-hand sides flank the sign; setting the right side to zero loses no generality.
Polynomial equations are the most common type, with terms on each side. Unknowns like x and y contrast with parameters such as A, B, and C whose roles context fixes. Solving finds values making the equality true; results may be specific numbers or expressions in parameters. Systems demand values satisfying every equation simultaneously.
Equations behave like balanced scales: equal weights on both pans stay level, and removing the same amount from each side preserves balance. Equivalent transformations include adding or subtracting the same quantity, multiplying or dividing by a nonzero value, applying identities, and combining equations in systems.
Applying functions to both sides can introduce extraneous solutions. Squaring x = 3 yields x² = 9, which also accepts x = −3. Operations undefined at some values, like dividing by x when x = 0, may discard valid roots. Identities such as (x + 1)² = x² + 2x + 1 simplify solving across algebra and trigonometry. A general quadratic ax² + bx + c = 0 typically reserves x, y, and z for unknowns and a, b, c for coefficients. Parameters can make one equation stand for a whole family: with R left open, x² + y² = R² describes every circle around the origin, and fixing R = 2 picks out one of radius 2. Trigonometric identities earn their keep too—rewriting 3 sin θ cos θ = 1 with the double-angle rule gives θ of roughly 20.9° once the angle is limited to between 0 and 45 degrees. The same equivalence moves also drive Gaussian elimination for larger systems.
Source: Equation
