Editing Quadratic equation (section)

==Examples and applications==
[[File:La Jolla Cove cliff diving - 02.jpg|thumb|The trajectory of the cliff jumper is [[parabola|parabolic]] because horizontal displacement is a linear function of time <math>x=v_x t</math>, while vertical displacement is a quadratic function of time <math>y=\tfrac{1}{2} at^2+v_y t+h</math>. As a result, the path follows quadratic equation <math>y=\tfrac{a}{2v_x^2} x^2+\tfrac{v_y}{v_x} x+h</math>, where <math>v_x</math> and <math>v_y</math> are horizontal and vertical components of the original velocity, {{math|a}} is [[Gravity of Earth|gravitational]] [[acceleration]] and {{math|h}} is original height. The {{math|a}} value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).]]
The [[golden ratio]] is found as the positive solution of the quadratic equation <math>x^2-x-1=0.</math>

The equations of the [[circle]] and the other [[conic sections]]&mdash;[[ellipse]]s, [[parabola]]s, and [[hyperbola]]s&mdash;are quadratic equations in two variables.

Given the [[cosine]] or [[sine]] of an angle, finding the cosine or sine of [[Bisection#Angle bisector|the angle that is half as large]] involves solving a quadratic equation.

The process of simplifying expressions involving the [[nested radical|square root of an expression involving the square root of another expression]] involves finding the two solutions of a quadratic equation.

[[Descartes' theorem]] states that for every four kissing (mutually tangent) circles, their [[radius|radii]] satisfy a particular quadratic equation.

The equation given by [[Fuss' theorem]], giving the relation among the radius of a [[bicentric quadrilateral]]'s [[inscribed circle]], the radius of its [[circumscribed circle]], and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the [[excircle]] of an [[ex-tangential quadrilateral]].

[[Critical point (mathematics)|Critical points]] of a [[cubic function]] and [[inflection point]]s of a [[quartic function]] are found by solving a quadratic equation.

In [[physics]], for [[motion]] with constant [[acceleration]] <math>a</math>, the [[Displacement (geometry)|displacement]] or position <math>x</math> of a moving body can be expressed as a [[quadratic function]] of [[time]] <math>t</math> given the initial position <math>x_0</math> and initial [[velocity]] <math>v_0</math>: <math display="inline">x = x_0 + v_0 t + \frac{1}2 at^2</math>.

In [[chemistry]], the [[pH]] of a [[Solution (chemistry)|solution]] of [[Acid strength#Weak acids|weak acid]] can be calculated from the negative [[Common logarithm|base-10 logarithm]] of the positive root of a quadratic equation in terms of the [[Acid dissociation constant|acidity constant]] and the [[Molar concentration#Formality or analytical concentration|analytical concentration]] of the acid.