Quadratic equation
In
The values of x that satisfy the equation are called
Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.[4][5]
Because the quadratic equation involves only one unknown, it is called "
Solving the quadratic equation
A quadratic equation with
Factoring by inspection
It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. 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. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if px + q = 0 or rx + s = 0. Solving these two linear equations provides the roots of the quadratic.
For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.[6]: 202–207 If one is given a quadratic equation in the form x2 + bx + c = 0, the sought factorization has the form (x + q)(x + s), and one has to find two numbers q and s that add up to b and whose product is c (this is sometimes called "Vieta's rule"[7] and is related to Vieta's formulas). As an example, x2 + 5x + 6 factors as (x + 3)(x + 2). 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.
Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.[6]: 207
Completing the square
The process of completing the square makes use of the algebraic identity
which represents a well-defined algorithm that can be used to solve any quadratic equation.[6]: 207 Starting with a quadratic equation in standard form, ax2 + bx + c = 0
- Divide each side by a, the coefficient of the squared term.
- Subtract the constant term c/a from both sides.
- Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square.
- Write the left side as a square and simplify the right side if necessary.
- Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
- Solve each of the two linear equations.
We illustrate use of this algorithm by solving 2x2 + 4x − 4 = 0
The plus–minus symbol "±" indicates that both x = −1 + √3 and x = −1 − √3 are solutions of the quadratic equation.[8]
Quadratic formula and its derivation
Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula.[9] The mathematical proof will now be briefly summarized.[10] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation:
Taking the square root of both sides, and isolating x, gives:
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 result in slightly different forms for the solution, but are otherwise equivalent.
A number of alternative derivations can be found in the literature. 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.
A lesser known quadratic formula, as used in Muller's method, provides the same roots via the equation
This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a. It also follows from dividing the quadratic equation by giving solving this for and then inverting.
One property of this form is that it yields one valid root when a = 0, while the other root contains division by zero, because when a = 0, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0.
When neither a nor c is zero, the equality between the standard quadratic formula and Muller's method,
can be verified by
Reduced quadratic equation
It is sometimes convenient to reduce a quadratic equation so that its
where p = b/a and q = c/a. This monic polynomial equation has the same solutions as the original.
The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is
or, equivalently,
Discriminant
In the quadratic formula, the expression underneath the square root sign is called the discriminant of the quadratic equation, and is often represented using an upper case D or an upper case Greek delta:[13]
A quadratic equation with real coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:
- If the discriminant is positive, then there are two distinct roots
- both of which are real numbers. For quadratic equations with quadratic irrationals.
- If the discriminant is zero, then there is exactly one real root sometimes called a repeated or double rootor two equal roots.
- If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) complex roots[14]
- which are complex conjugates of each other. In these expressions i is the imaginary unit.
Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.
Geometric interpretation
The function f(x) = ax2 + 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. If a < 0, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its vertex. The x-coordinate of the vertex will be located at , and the y-coordinate of the vertex may be found by substituting this x-value into the function. The y-intercept is located at the point (0, c).
The solutions of the quadratic equation ax2 + bx + c = 0 correspond to the
Quadratic factorization
The term
is a factor of the polynomial
if and only if r is a
It follows from the quadratic formula that
In the special case b2 = 4ac where the quadratic has only one distinct root (i.e. the discriminant is zero), the quadratic polynomial can be factored as
Graphical solution
The solutions of the quadratic equation
may be deduced from the graph of the quadratic function
which is a parabola.
If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept).
If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola.
If the parabola does not intersect the x-axis, there are two complex conjugate roots. Although these roots cannot be visualized on the graph, their real and imaginary parts can be.[17]
Let h and k be respectively the x-coordinate and the y-coordinate of the vertex of the parabola (that is the point with maximal or minimal y-coordinate. The quadratic function may be rewritten
Let d be the distance between the point of y-coordinate 2k on the axis of the parabola, and a point on the parabola with the same y-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is h, and their imaginary part are ±d. That is, the roots are
or in the case of the example of the figure
Avoiding loss of significance
Although the quadratic formula provides an exact solution, the result is not exact if
This occurs when the roots have different
using the plus sign if and the minus sign if
A second form of cancellation can occur between the terms b2 and 4ac of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.[11][18]
Examples and applications
The golden ratio is found as the positive solution of the quadratic equation
The equations of the
—are quadratic equations in two variables.Given the
The process of simplifying expressions involving the 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 radii satisfy a particular quadratic equation.
The equation given by
Critical points of a cubic function and inflection points of a quartic function are found by solving a quadratic equation.
History
which is equivalent to the statement that x and y are the roots of the equation:[20]: 86
The steps given by Babylonian scribes for solving the above rectangle problem, in terms of x and y, were as follows:
- Compute half of p.
- Square the result.
- Subtract q.
- Find the (positive) square root using a table of squares.
- Add together the results of steps (1) and (4) to give x.
In modern notation this means calculating , which is equivalent to the modern day quadratic formula for the larger real root (if any) with a = 1, b = −p, and c = q.
Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.[21] Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots.[22][23] Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.[23][24] These early geometric methods do not appear to have had a general formula. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive.[25]
In 628 AD, Brahmagupta, an Indian mathematician, gave in his book Brāhmasphuṭasiddhānta the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."[26] This is equivalent to
The
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.[32] His solution was largely based on Al-Khwarizmi's work.[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.[33] By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594.[34] In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.
Advanced topics
Alternative methods of root calculation
Vieta's formulas
Vieta's formulas (named after François Viète) are the relations
between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation
with the equation
The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's x-coordinate is located at the average of the roots (or intercepts). Thus the x-coordinate of the vertex is
The y-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
Also, these formulas for the vertex can be deduced directly from the formula (see Completing the square)
For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If |x2| << |x1|, then x1 + x2 ≈ x1, and we have the estimate:
The second Vieta's formula then provides:
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. 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). As the linear coefficient b increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as b increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.
This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response).
Trigonometric solution
In the days before calculators, people would use mathematical tables—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. 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. Consider the following alternate form of the quadratic equation,
[1]
where the sign of the ± symbol is chosen so that a and c may both be positive. By substituting
[2]
and then multiplying through by cos2(θ) / c, we obtain
[3]
Introducing functions of 2θ and rearranging, we obtain
[4]
[5]
where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation [1]. 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. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone.[36] Calculating complex roots would require using a different trigonometric form.[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:
- A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries.
- (rounded to six significant figures)
Solution for complex roots in polar coordinates
If the quadratic equation with real coefficients has two complex roots—the case where requiring a and c to have the same sign as each other—then the solutions for the roots can be expressed in polar form as[38]
where and
Geometric solution
The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure 6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA.[39]
The
Generalization of quadratic equation
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. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.)
The symbol
in the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Even if a field does not contain a square root of some number, there is always a quadratic
Characteristic 2
In a field of characteristic 2, the quadratic formula, which relies on 2 being a unit, does not hold. Consider the monic quadratic polynomial
over a field of characteristic 2. If b = 0, then the solution reduces to extracting a square root, so the solution is
and there is only one root since
In summary,
See quadratic residue for more information about extracting square roots in finite fields.
In the case that b ≠ 0, there are two distinct roots, but if the polynomial is irreducible, they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the 2-root R(c) of c to be a root of the polynomial x2 + x + c, an element of the splitting field of that polynomial. One verifies that R(c) + 1 is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic ax2 + bx + c are
and
For example, let a denote a multiplicative generator of the group of units of F4, the
This is a special case of Artin–Schreier theory.
See also
- Solving quadratic equations with continued fractions
- Linear equation
- Cubic function
- Quartic equation
- Quintic equation
- Fundamental theorem of algebra
References
- ^ Protters & Morrey: "Calculus and Analytic Geometry. First Course".
- ^ ISBN 978-0-201-35666-3.
- ISBN 9780387974972.
- ISBN 978-0-470-55964-2
- ^ Himonas, Alex. Calculus for Business and Social Sciences, p. 64 (Richard Dennis Publications, 2001).
- ^ a b Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic (PDF), retrieved 2012-12-25
- ^ Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. Concise Handbook of Mathematics and Physics, p. 38 (CRC Press 1997)
- ^ Δ is the initial of the Greek word Διακρίνουσα, Diakrínousa, discriminant.
- ISBN 978-0-8311-3086-2.
- ^ "Complex Roots Made Visible – Math Fun Facts". Retrieved 1 October 2016.
- ISBN 978-1-905-129-78-2.
- JSTOR 2686333
- ISBN 978-0-89871-521-7
- ^ Friberg, Jöran (2009). "A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma". Cuneiform Digital Library Journal. 3.
- ISBN 978-0-387-95336-6.
- ISBN 978-0-521-07791-0.
- ^ Henderson, David W. "Geometric Solutions of Quadratic and Cubic Equations". Mathematics Department, Cornell University. Retrieved 28 April 2013.
- ^ a b Aitken, Wayne. "A Chinese Classic: The Nine Chapters" (PDF). Mathematics Department, California State University. Retrieved 28 April 2013.
- ISBN 978-0-486-20430-7.
- ISBN 978-0-387-95336-6.
- ^ S2CID 120363574.
- ^ ISBN 978-0-471-54397-8.
- ^ O'Connor, John J.; Robertson, Edmund F. (1999), "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics Archive, University of St Andrews "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."
- ISBN 978-1-4020-0260-1
- ISBN 978-0-486-20429-1.
- ISBN 978-0743258210.
- ISBN 978-0-521-31536-4.
- ^ Struik, D. J.; Stevin, Simon (1958), The Principal Works of Simon Stevin, Mathematics (PDF), vol. II–B, C. V. Swets & Zeitlinger, p. 470
- ^ Ballew, Pat. "Solving Quadratic Equations — By analytic and graphic methods; Including several methods you may never have seen" (PDF). Archived from the original (PDF) on 9 April 2011. Retrieved 18 April 2013.
- doi:10.1086/125759.
- JSTOR 3028750.
- ^ Simons, Stuart, "Alternative approach to complex roots of real quadratic equations", Mathematical Gazette 93, March 2009, 91–92.
- ^ Bixby, William Herbert (1879), Graphical Method for finding readily the Real Roots of Numerical Equations of Any Degree, West Point N. Y.
- ^ Weisstein, Eric W. "Carlyle Circle". From MathWorld—A Wolfram Web Resource. Retrieved 21 May 2013.
External links
- "Quadratic equation", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric W. "Quadratic equations". MathWorld.
- 101 uses of a quadratic equation Archived 2007-11-10 at the Wayback Machine
- 101 uses of a quadratic equation: Part II Archived 2007-10-22 at the Wayback Machine