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Because (a + 1) 2 = a, a + 1 is the unique solution of the quadratic equation x 2 + a = 0. On the other hand, the polynomial x 2 + ax + 1 is irreducible over F 4 , but it splits over F 16 , where it has the two roots ab and ab + a , where b is a root of x 2 + x + a in F 16 .
This crucial step completes a larger square of side length + . Completing the square is the oldest method of solving general quadratic equations, used in Old Babylonian clay tablets dating from 1800–1600 BCE
The quadratic equation on a number can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated ...
A similar but more complicated method works for cubic equations, which have three resolvents and a quadratic equation (the "resolving polynomial") relating and , 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. [14]
Photomath utilizes the camera of a user's smartphone or tablet to scan and identify mathematical problems. [4] Upon recognition, the app displays the steps to solve the problem. The app presents these steps through various methods and approaches, elucidating the problem-solving process in a step-by-step manner to educate users.
For example, the equation x + y = 2x – 1 is solved for the unknown x by the expression x = y + 1, because substituting y + 1 for x in the equation results in (y + 1) + y = 2(y + 1) – 1, a true statement. It is also possible to take the variable y to be the unknown, and then the equation is solved by y = x – 1.
The matrix equation AX=XB, where X is unknown, has an infinitive number of solutions that can be easily studied by a geometrical approach. [8] To find X it is necessary to consider a simultaneous set of 2 equations A 1 X=XB 1 and A 2 X=XB 2; the matrices A 1, A 2, B 1, B 2 have to be dermined by experiments to be performed in an optimized way. [9]
The consequence of this difference is that at every step, a system of algebraic equations has to be solved. This increases the computational cost considerably. If a method with s stages is used to solve a differential equation with m components, then the system of algebraic equations has ms components.
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