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For quadratic equations with rational coefficients, if the discriminant is a square number, then the roots are rational—in other cases they may be quadratic irrationals. If the discriminant is zero, then there is exactly one real root − b 2 a , {\displaystyle -{\frac {b}{2a}},} sometimes called a repeated or double root or two equal roots.
Geometrically, the discriminant of a quadratic form in three variables is the equation of a quadratic projective curve. The discriminant is zero if and only if the curve is decomposed in lines (possibly over an algebraically closed extension of the field). A quadratic form in four variables is the equation of a projective surface.
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. [14]
which is a simple quadratic equation, whose solutions are easily found using the quadratic formula: ... This can be done by letting the discriminant of the quadratic ...
He was the first who discovered the rules for summing the powers of the roots of any equation. In this vein, the discriminant is a symmetric function in the roots that reflects properties of the roots – it is zero if and only if the polynomial has a multiple root, and for quadratic and cubic polynomials it is positive if and only if all roots ...
If a quadratic function is equated with zero, then the result is a quadratic equation. The solutions of a quadratic equation are the zeros (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the quadratic formula. A quadratic polynomial or quadratic function can involve ...
A former Florida sheriff's deputy has been identified as the suspect in a store clerk's murder that took place 45 years ago, the Charlotte County Sheriff's Office said. John Greer has been named ...
This follows directly from the quadratic formula: = . where, if the discriminant b 2 −4ac is less than zero, then the polynomial will have two complex-conjugate solutions with real part −b/2a, which is negative for positive a and b.