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The discriminant B 2 – 4AC of the conic section's quadratic equation (or equivalently the determinant AC – B 2 /4 of the 2 × 2 matrix) and the quantity A + C (the trace of the 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, [14] [15] [16] as is the determinant of the 3 × 3 matrix above.
If the discriminant is negative, then the parabola opens in the opposite direction, never crossing the -axis, and the equation has no real roots; in this case the two complex-valued roots will be complex conjugates whose real part is the value of the axis of symmetry.
In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic ...
Graph of y = ax 2 + bx + c, where a and the discriminant b 2 − 4ac are positive, with. Roots and y-intercept in red; Vertex and axis of symmetry in blue; Focus and directrix in pink; Visualisation of the complex roots of y = ax 2 + bx + c: the parabola is rotated 180° about its vertex (orange).
However, the discriminant in a PDE is given by B 2 − AC due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B) 2 − 4AC = 4(B 2 − AC), with the factor of 4 dropped for simplicity.
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. If the discriminant is equal to zero, there will be two coinciding real solutions at −b/2a.
Consider a quadratic form given by f(x,y) = ax 2 + bxy + cy 2 and suppose that its discriminant is fixed, say equal to −1/4. In other words, b 2 − 4ac = 1. One can ask for the minimal value achieved by | (,) | when it is evaluated at non-zero vectors of the grid , and if this minimum does not exist, for the infimum.
The discriminant of a polynomial equation, especially the quadratic equation: [7] [8] Δ = b 2 − 4 a c . {\displaystyle \Delta =b^{2}-4ac.} The area of a triangle