Using the Quadratic Formula http://www.gandraxa.com/using_the_quadratic_formula.xml?Var1=3;Var2=1/4;Var3=4 img/uqf_menu.jpg Cannonballs have trajectories expressed by quadratic formulas img/uqf.jpg home.xml Home articles.xml Articles Using the Quadratic Formula herbert gandraxa.com Herbert Glarner 2011Jun02 First published. http://en.wikipedia.org/wiki/Quadratic_equation Quadratic equation on Wikipedia http://www.purplemath.com/modules/sqrquad2.htm Deriving the formula on Purplemath http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-4/ The Motion of Projectiles , video lecture 4 of MIT course 18.01 Physics I (in which Prof. Walter Lewin shoots a monkey).

Abstract Demonstrates how the solutions of a quadratic equation can be found by using the quadratic formula.

When a polynomial equation of second degree (quadratic equation) is given in the form

$ax2+ bx + c = 0$

then this page will find the up to 2 real or complex solutions x1 and x2 (called roots) via the quadratic formula

$x$1, x2 =nbsp -b plusmnnbsp b2 - 4ac 2a
Instructions

Enter the 3 coefficients `a`, `b` and `c`. This page will then find the up to 2 real or complex roots.

• Only integers and well-formed fractions will be accepted. A denominator of 0 will return an error.
• Use fractions, not decimals (enter `1/4`, not `0.25`)
• Don't use mixed numbers (enter `5/3`, not `1 2/3`)
• When negative, use the minus sign before the numerator (enter `-1/3`, not `1/-3`)
• All individual numbers (numerator, denominator) may not exceed 6 digits: this should be enough to demonstrate how the formula is used. However, even if you make sure, that your input stays within this limit, any intermediate calculation may still exceed the internal maximum number length (28 digits), especially when working with fractions, eventhough it uses the visualising_the_euclidean_algorithm.xml Euclidean algorithm to keep fractions as small as possible. You might also experience a timeout, when the server decides that you consumed enough of its time.
• Input

Enter the 3 coefficients a, b and c, where a ne 0 (with a = 0 the quadratic equation would become a linear equation):

a = , b = , c = .
Solutions

With a = 3, b = 1/4, c = 4, we have the quadratic equation:

$-1/4 plusmnnbsp(1/4)2 - 4times3times42times3$$-1/4 plusmnnbsp1/16 - 486$$-1/4 plusmnnbsp-767/166$

The term underneath the root, b2 - 4ac, is called Discriminant. Its value governs, how many solutions we get, and of what type these solutions are. dash Here, the discriminant has a value of -767/16. Whenever the discriminant is not 0, there are two solutions, and because the value is negative, both solutions are complex.

Simplifying the root by integer factorization, if possible (using the http://en.wikipedia.org/wiki/Sieve_of_EratosthenesSieve of Eratosthenes):$-13 times 5924nbsp=nbsp14i767$

Plugging the discriminant back into the formula:

$-1/4nbspplusmnnbsp(1/4)i7676$

Calculating the formula's simplest terms:

$-1/24nbspplusmnnbsp(1/24)i767$

Hence the two solutions are:

$-1/24nbsp+nbsp(1/24)i767nbsp (decimal approx. -0,0416666666666667 + 1,15394853534385$i)

and

$-1/24nbsp-nbsp(1/24)i767nbsp (decimal approx. -0,0416666666666667 - 1,15394853534385$i)