Search results
Results from the WOW.Com Content Network
The four quadrants of a Cartesian coordinate system The axes of a two-dimensional Cartesian system divide the plane into four infinite regions , called quadrants , each bounded by two half-axes. The axes themselves are, in general, not part of the respective quadrants.
For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x 2 + y 2 = 4; the area, the perimeter and the tangent line at any point can be computed from this equation by using integrals and derivatives, in a way that can be applied to any curve.
Some have come up with advanced and complex algebraic or trigonometric equations to approximate the heart/love symbol. This image instead uses two simple perpendicular straight lines and two partial circular arcs. Mathematically, x and y are related by the following four equations (one applying in each quadrant): Ist quadrant (x≥0,y≥0)
The horizontal plane shows the four quadrants between x- and y-axis. (Vertex numbers are little-endian balanced ternary.) An octant in solid geometry is one of the eight divisions of a Euclidean three-dimensional coordinate system defined by the signs of the coordinates. It is analogous to the two-dimensional quadrant and the one-dimensional ...
The four roots of the depressed quartic x 4 + px 2 + qx + r = 0 may also be expressed as the x coordinates of the intersections of the two quadratic equations y 2 + py + qx + r = 0 and y − x 2 = 0 i.e., using the substitution y = x 2 that two quadratics intersect in four points is an instance of Bézout's theorem.
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]
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.
For example, the equation z 2 + 1 = 0, has infinitely many quaternion solutions, which are the quaternions z = b i + c j + d k such that b 2 + c 2 + d 2 = 1. Thus these "roots of –1" form a unit sphere in the three-dimensional space of vector quaternions.