Theory of equations/Linear and quadratic equations

Linear equations Quadratic equations Type Basic form Solution method Number of solutions Standard form Discriminant Nature of roots Real coefficients (with ) Isolate : Exactly one real solution : two distinct real roots. : one real double root. : two complex conjugate roots. Homogeneous Either (if ) or any (if ) One or infinitely many solutions If : roots and If : reduces to linear or trivial equation Complex coefficients , Exactly one complex solution , Always two (not necessarily distinct) complex roots Linear Quadratic Standard form Degree 1 2 Typical solution formula Graph (real case) Straight line Parabola Maximum number of real roots 1 2