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Quadratic formula. The roots of the quadratic function y = 1 2 x2 − 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.
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as [1] + + =, where x represents an unknown value, and a, b, and c represent known numbers, where a ≠ 0.
For any positive integer m and any non-negative integer n, the multinomial theorem describes how a sum with m terms expands when raised to the n th power: where is a multinomial coefficient. This can be proved by the slider method. The sum is taken over all combinations of nonnegative integer indices k1 through km such that the sum of all ki is n.
Polynomial expansion. In mathematics, an expansion of a product of sums expresses it as a sum of products by using the fact that multiplication distributes over addition. Expansion of a polynomial expression can be obtained by repeatedly replacing subexpressions that multiply two other subexpressions, at least one of which is an addition, by ...
The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that that is, the characteristic polynomial is an annihilating polynomial for. One use for the Cayley–Hamilton theorem is that it allows An to be expressed as a linear combination of the lower matrix powers of A: When the ring is ...
Identity 1: The following two results follow from this and the ratio identities. To obtain the first, divide both sides of by ; for the second, divide by . Similarly. Identity 2: The following accounts for all three reciprocal functions. Proof 2: Refer to the triangle diagram above. Note that by Pythagorean theorem.
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial.According to the theorem, it is possible to expand the polynomial (x + y) n into a sum involving terms of the form ax b y c, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending ...
v. t. e. Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number x, one has where e is the base of the natural logarithm, i is the imaginary ...