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The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola. If a > 0 , {\displaystyle a>0,} then the equation y = ± a x 2 + b x + c {\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}} describes a hyperbola, as can be seen by squaring both sides.
The function f(x) = ax 2 + bx + c is a quadratic function. [16] The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of a, b, and c. If a > 0, the parabola has a minimum point and opens upward.
The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
A closely related notion with geometric overtones is a quadratic space, which is a pair (V, q), with V a vector space over a field K, and q : V → K a quadratic form on V. See § Definitions below for the definition of a quadratic form on a vector space.
In elementary algebra, completing the square is a technique for converting a quadratic polynomial of the form + + to the form + for some values of and . [1] In terms of a new quantity x − h {\displaystyle x-h} , this expression is a quadratic polynomial with no linear term.
By definition, a quadric X of dimension n over a field k is the subspace of + defined by q = 0, where q is a nonzero homogeneous polynomial of degree 2 over k in variables , …, +. (A homogeneous polynomial is also called a form, and so q may be called a quadratic form.)
For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets i for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets −i. There are generally two ways of solving the problem.
Quadratic formula, calculation to solve a quadratic equation for the independent variable (x) Quadratic field, an algebraic number field of degree two over the field of rational numbers; Quadratic irrational or "quadratic surd", an irrational number that is a root of a quadratic polynomial
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