Question

If the discriminant is 13, how many solutions and of what type?

Answer 1

A polynomial's discriminant is a number that depends on the coefficients and defines different features of the roots in mathematics. It's usually described as a polynomial function of the original polynomial's coefficients. In polynomial factoring, number theory, and algebraic geometry, the discriminant is frequently used. The symbol delta is frequently used to represent it.

Complete answer:
The discriminant of the given quadratic polynomial $a{x^2} + bx + c{\mkern 1mu}$ is $\Delta = {b^2} - 4ac$ the quantity in the quadratic formula that occurs after the square root. If and only if the polynomial has a double root, this discriminant is zero. If the polynomial has two separate real roots, it is positive; if the polynomial has two distinct complex conjugate roots, it is negative. Similarly, there is a discriminant for a cubic polynomial that is zero if and only if the polynomial has many roots. If a cubic with real coefficients has three different real roots, the discriminant is positive; if it has one real root and two unique complex conjugate roots, it is negative. In general, the discriminant of a positive degree univariate polynomial is zero if and only if the polynomial has many roots. The discriminant is positive for real coefficients and no multiple roots if the number of non-real roots is a multiple of 4 (including none), and negative otherwise.
The discriminant in this case is 13.
Because the discriminant is not equal to zero, it is positive.
Real and distinct solutions will be provided.
There are two possible roots.

Note:
The discriminant of an algebraic number field, the discriminant of a quadratic form, and, more broadly, the discriminant of a form, a homogeneous polynomial, or a projective hypersurface are all examples of generalisations (these three concepts are essentially equivalent).