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The roots of the quadratic function y = 1 / 2 x 2 − 3x + 5 / 2 are the places where the graph intersects the x-axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
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.
For example, taking the statement x + 1 = 0, if x is substituted with 1, this implies 1 + 1 = 2 = 0, which is false, which implies that if x + 1 = 0 then x cannot be 1. If x and y are integers, rationals, or real numbers, then xy = 0 implies x = 0 or y = 0. Consider abc = 0. Then, substituting a for x and bc for y, we learn a = 0 or bc = 0.
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 is a special case of Artin–Schreier theory.
When solving a system over a finite field k with q elements, one is primarily interested in the solutions in k. As the elements of k are exactly the solutions of the equation x q – x = 0, it suffices, for restricting the solutions to k, to add the equation x i q – x i = 0 for each variable x i.
2.3 Trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions relationship. 2.4 Modified-factorial denominators. 2.5 Binomial coefficients.
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Brahmagupta solved many Pell's equations with this method, proving that it gives solutions starting from an integer solution of = for k = ±1, ±2, or ±4. [11] The first general method for solving the Pell's equation (for all N) was given by Bhāskara II in 1150, extending the methods of Brahmagupta.