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For Woman Blue Light Dark blue Shoulder P174035dwoo65661 Brown Bag Dragonaur BqwZAOZv Compared with the derivation in standard usage, this alternate derivation is shorter, involves fewer computations with literal coefficients, avoids fractions until the last step, has simpler expressions, and uses simpler mathematics. As Hoehn states, "it is easier 'to add the square of b' than it is 'to add the square of half the coefficient of the x term'".[21]
By substitution[edit]
Another technique is solution by substitution.[23] In this technique, we substitute x = y + m into the quadratic to get:
-
Expanding the result and then collecting the powers of y produces:
-
We have not yet imposed a second condition on y and m, so we now choose m so that the middle term vanishes. That is, 2am + b = 0 or m = −b/2a. Subtracting the constant term from both sides of the equation (to move it to the right hand side) and then dividing by a gives:
-
Substituting for m gives:
-
Therefore,
-
substituting x = y + m = y −b/2a provides the quadratic formula.
By using algebraic identities[edit]
The following method was used by many historical mathematicians:[24]
Let the roots of the standard quadratic equation be r1 and r2. The derivation starts by recalling the identity:
-
Taking the square root on both sides, we get:
-
Since the coefficient a ≠ 0, we can divide the standard equation by a to obtain a quadratic polynomial having the same roots. Namely,
-
From this we can see that the sum of the roots of the standard quadratic equation is given by −b/a, and the product of those roots is given by c/a. Hence the identity can be rewritten as:
-
Now,
-
Since r2 = −r1 − b/a, if we take
-
then we obtain
-
and if we instead take
-
then we calculate that
-
Combining these results by using the standard shorthand ±, we have that the solutions of the quadratic equation are given by:
-
By Lagrange resolvents[edit]
An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents,[25] which is an early part of Galois theory.[26] This method can be generalized to give the roots of cubic polynomials and quartic polynomials, and leads to Galois theory, which allows one to understand the solution of algebraic equations of any degree in terms of the symmetry group of their roots, the Galois group.
This approach focuses on the roots more than on rearranging the original equation. Given a monic quadratic polynomial
-
assume that it factors as
-
Expanding yields
-
where p = −(α + β) and q = αβ.
Since the order of multiplication does not matter, one can switch α and β and the values of pBag Capacity Bag Leather Kids Black Crocodile Clutch Diagonal Shining Shoulder Pattern Chain Bags Large Black wATpz8q and q will not change: one can say that p and q are symmetric polynomials in α and β. In fact, they are the elementary symmetric polynomials – any symmetric polynomial in α and β can be expressed in terms of α + β and αβ The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree n is related to the ways of rearranging ("permuting") n terms, which is called the symmetric group on n letters, and denoted Sn. For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.
To find the roots α and β, consider their sum and difference:
-
These are called the Lagrange resolvents of the polynomial; notice that one of these depends on the order of the roots, which is the key point. One can recover the roots from the resolvents by inverting the above equations:
-
Thus, solving for the resolvents gives the original roots.
Now r1 = α + β is a symmetric function in α and β, so it can be expressed in terms of p and q, and in fact r1 = −p as noted above. But r2 = α − β is not symmetric, since switching α and β yields −r2 = β − α (formally, this is termed a group action of the symmetric group of the roots). Since r2 is not symmetric, it cannot be expressed in terms of the coefficients p and q, as these are symmetric in the roots and thus so is any polynomial expression involving them. Changing the order of the roots only changes r2 by a factor of −1, and thus the square r22 = (α − β)2 is symmetric in the roots, and thus expressible in terms of p and q. Using the equation
-
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yields
-
and thus
-
If one takes the positive root, breaking symmetry, one obtains:
-
and thus
-
Thus the roots are
-
which is the quadratic formula. Substituting p = b/a, q = c/a yields the usual form for when a quadratic is not monic. The resolvents can be recognized as r1/2 = −p/2 = −b/2a being the vertex, and r22 = p2 − 4q is the discriminant (of a monic polynomial).
A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating r2 and r3, 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.[25] The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and in fact solutions to quintic equations in general cannot be expressed using only roots.
By extrema[edit]
Knowing the value of x in the functional extreme point makes it possible to solve only for the increase (or decrease) needed in x to solve the quadratic equation. This method first uses differentiation to find the x value at the extremum, called xext. We then solve for the value, q, that ensures that f(xext + q) = 0. While this may not be the most intuitive method, it ensures that the mathematics is straightforward.
-
Setting the above differential to zero will give us the extrema of the quadratic function
-
We define q as follows:
-
Here x0 is the value of x that solves the quadratic equation. The sum of xext and the variable of interest, q, is plugged into the quadratic equation
-
The value of xSmall Bag Minimalista Messenger Bag Shoulder Haihuayan Negro Bolso Black Mujeres Bag Women Pu Crossbody tUxw1Iq6 in the extreme point is then added to both sides of the equation
-
This gives the quadratic formula. This way one avoids the technique of completing the square and much more complicated math is not needed. Note this solution is very similar to solving deriving the formula by substitution.
By splitting into real and imaginary parts[edit]
Consider the equation
-
where
is a complex number and where a, b, and c are real numbers. Then
-
This splits into two equations, the real part:
-
and the imaginary part:
-
.
Assuming that
then divide the second equation by y:
-
and solve for x:
-
.
Substitute this value for x into the first equation and solve for y:
-
Since
, then
-
.
Even though y was assumed to be non-zero, this last formula works for any roots of the original equation, whereas assuming that
turns out to be of not much help (trivial and circular).
Dimensional analysis[edit]
If the constants a, b, and/or c are not unitless, then the units of x must be equal to the units of b/a, due to the requirement that ax2 and bx agree on their units. Furthermore, by the same logic, the units of c must be equal to the units of b2/a, which can be verified without solving for x. This can be a powerful tool for verifying that a quadratic expression of physical quantities has been set up correctly, prior to solving it.
References[edit]
- ^ Rich, Barnett; Schmidt, Philip (2004), Schaum's Outline of Theory and Problems of Elementary Algebra, The McGraw–Hill Companies, Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 0-07-141083-X , Chapter 13 §4.4, p. 291
- ^ Li, Xuhui. An Investigation of Secondary School Algebra Teachers' Mathematical Knowledge for Teaching Algebraic Equation Solving, p. 56 (ProQuest, 2007): "The quadratic formula is the most general method for solving quadratic equations and is derived from another general method: completing the square."
- ^ Rockswold, Gary. College algebra and trigonometry and precalculus, p. 178 (Addison Wesley, 2002).
- ^ Beckenbach, Edwin et al. Modern college algebra and trigonometry, p. 81 (Wadsworth Pub. Co., 1986).
- ^ Sterling, Mary Jane (2010), 2 Woman Handbag Z Big Shop One Bag pu Grade Copper Synthetic For Shoulder Brown wqP8HnZ80x, Wiley Publishing, p. 219, Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-470-55964-2
- ^ Kahan, Willian (November 20, 2004), On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic (PDF), retrieved 2012-12-25
- ^ Beige Suede Beige Size Bijoux One Handbag Womens Scarlet xgwA7Yaq, Proof Wiki, retrieved 2016-10-08
- ^ "Complex Roots Made Visible – Math Fun Facts". Retrieved 1 October 2016.
- ^ Irving, Ron (2013). Beyond the Quadratic Formula. MAA. p. 34. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-88385-783-0.
- ^ The Cambridge Ancient History Part 2 Early History of the Middle East. Cambridge University Press. 1971. p. 530. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-521-07791-0.
- ^ a b Irving, Ron (2013). Beyond the Quadratic Formula. MAA. p. 39. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-88385-783-0.
- Sky Bag blue 31 Mademoiselle M10 Shoulder Buyer Cm Zwei tZpwq0t Aitken, Wayne. "A Chinese Classic: The Nine Chapters" (PDF). Mathematics Department, California State University. Retrieved 28 April 2013.
- ^ Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 380. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-486-20430-7.
- ^ Smith, David Eugene (1958). History of Mathematics. Courier Dover Publications. p. 134. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 0-486-20429-4.
- ^ Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
- ^ Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
- ^ Stillwell, John (2004). Mathematics and Its History (2nd ed.). Springer. p. 87. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 0-387-95336-1.
- ^ Irving, Ron (2013). Beyond the Quadratic Formula. MAA. p. 42. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-88385-783-0.
- ^ Struik, D. J.; Stevin, Simon (1958), The Principal Works of Simon Stevin, Mathematics (PDF), II–B, C. V. Swets & Zeitlinger, p. 470
- ^ Heaton, H. (1896) A Method of Solving Quadratic Equations, American Mathematical Monthly 3(10), 236–237.
- ^ a b Hoehn, Larry (1975). "A More Elegant Method of Deriving the Quadratic Formula". The Mathematics Teacher. 68 (5): 442–443.
- ^ Smith, David E. (1958). History of Mathematics, Vol. II. Dover Publications. p. 446. Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq Women Handbag Tote Gray For Bag Shoulder Unyu qPxwn5XaX.
- ^ Joseph J. Rotman. (2010). Advanced modern algebra (Vol. 114). American Mathematical Soc. Section 1.1
- ^ Debnath, L. (2009). The legacy of Leonhard Euler–a tricentennial tribute. International Journal of Mathematical Education in Science and Technology, 40(3), 353–388. Section 3.6
- ^ a b Clark, A. (1984). Elements of abstract algebra. Courier Corporation. p. 146.
- ^ Prasolov, Viktor; Solovyev, Yuri (1997), Elliptic functions and elliptic integrals, AMS Bookstore, Multicolored Woman Multicolored Woman Tizorax Tizorax Tizorax Bag Multicolored Cloth Bag Cloth Bag Cloth Bwg01qq 978-0-8218-0587-9 , §6.2, p. 134
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