Quadratic equation

A quadratic equation whose coefficients are real numbers can have either zero, one, or two distinct real-valued solutions, also called roots.

A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct.

In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another.

The more general case where a does not equal 1 can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.

[6]: 207  Starting with a quadratic equation in standard form, ax2 + bx + c = 0 We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 − 2bx + c = 0 ,[11] where b has a magnitude one half of the more common one, possibly with opposite sign.

These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root.

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

[19] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:

The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y, were as follows: In modern notation this means calculating

with a = 1, b = −p, and c = q. Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India.

With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation.

Muhammad ibn Musa al-Khwarizmi (9th century) developed a set of formulas that worked for positive solutions.

[27] He also described the method of completing the square and recognized that the discriminant must be positive,[27][28]: 230  which was proven by his contemporary 'Abd al-Hamīd ibn Turk (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.

[30] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.

[31] The Jewish mathematician Abraham bar Hiyya Ha-Nasi (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.

[27] The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.

[34] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.

The figure shows the difference between[clarification needed] (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced).

However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve.

This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response).

Methods of numerical approximation existed, called prosthaphaeresis, that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.

[35] Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in celestial mechanics calculations.

It is within this context that we may understand the development of means of solving quadratic equations by the aid of trigonometric substitution.

Substituting the two values of θn or θp found from equations [4] or [5] into [2] gives the required roots of [1].

Complex roots occur in the solution based on equation [5] if the absolute value of sin 2θp exceeds unity.

[37] To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy:

The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2.

Figure 1. Plots of the quadratic function, y = eh x squared plus b x plus c, varying each coefficient separately while the other coefficients are fixed at values eh = 1, b = 0, c = 0. The left plot illustrates varying c. When c equals 0, the vertex of the parabola representing the quadratic function is centered on the origin, and the parabola rises on both sides of the origin, opening to the top. When c is greater than zero, the parabola does not change in shape, but its vertex is raised above the origin. When c is less than zero, the vertex of the parabola is lowered below the origin. The center plot illustrates varying b. When b is less than zero, the parabola representing the quadratic function is unchanged in shape, but its vertex is shifted to the right of and below the origin. When b is greater than zero, its vertex is shifted to the left of and below the origin. The vertices of the family of curves created by varying b follow along a parabolic curve. The right plot illustrates varying eh. When eh is positive, the quadratic function is a parabola opening to the top. When eh is zero, the quadratic function is a horizontal straight line. When eh is negative, the quadratic function is a parabola opening to the bottom.
Figure 1. Plots of quadratic function y = ax 2 + bx + c , varying each coefficient separately while the other coefficients are fixed (at values a = 1, b = 0, c = 0)
Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals −1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.
Figure 2. For the quadratic function y = x 2 − x − 2 , the points where the graph crosses the x -axis, x = −1 and x = 2 , are the solutions of the quadratic equation x 2 − x − 2 = 0 .
Figure 3. This figure plots three quadratic functions on a single Cartesian plane graph to illustrate the effects of discriminant values. When the discriminant, delta, is positive, the parabola intersects the x-axis at two points. When delta is zero, the vertex of the parabola touches the x-axis at a single point. When delta is negative, the parabola does not intersect the x-axis at all.
Figure 3. Discriminant signs
Visualisation of the complex roots of y = ax 2 + bx + c : the parabola is rotated 180° about its vertex ( orange ). Its x -intercepts are rotated 90° around their mid-point, and the Cartesian plane is interpreted as the complex plane ( green ). [ 15 ]
Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation 2 x 2 + 4 x − 4 = 0 . Although the display shows only five significant figures of accuracy, the retrieved value of xc is 0.732050807569, accurate to twelve significant figures.
A quadratic function without real root: y = ( x − 5) 2 + 9 . The "3" is the imaginary part of the x -intercept. The real part is the x -coordinate of the vertex. Thus the roots are 5 ± 3 i .
The trajectory of the cliff jumper is parabolic because horizontal displacement is a linear function of time , while vertical displacement is a quadratic function of time . As a result, the path follows quadratic equation , where and are horizontal and vertical components of the original velocity, a is gravitational acceleration and h is original height. The a value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).
Figure 6. Geometric solution of eh x squared plus b x plus c = 0 using Lill's method. The geometric construction is as follows: Draw a trapezoid S Eh B C. Line S Eh of length eh is the vertical left side of the trapezoid. Line Eh B of length b is the horizontal bottom of the trapezoid. Line B C of length c is the vertical right side of the trapezoid. Line C S completes the trapezoid. From the midpoint of line C S, draw a circle passing through points C and S. Depending on the relative lengths of eh, b, and c, the circle may or may not intersect line Eh B. If it does, then the equation has a solution. If we call the intersection points X 1 and X 2, then the two solutions are given by negative Eh X 1 divided by S Eh, and negative Eh X 2 divided by S Eh.
Figure 6. Geometric solution of ax 2 + bx + c = 0 using Lill's method. Solutions are −AX1/SA, −AX2/SA
Carlyle circle of the quadratic equation x 2 − sx + p = 0.