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The quadratic formula can equivalently be written using various alternative expressions, for instance = (), which can be derived by first dividing a quadratic equation by , resulting in + + = , then substituting the new coefficients into the standard quadratic formula.
Symmetry: for every a and b, if a = b, then b = a. Transitivity: for every a, b, and c, if a = b and b = c, then a = c. [11] [12] Substitution: Informally, this just means that if a = b, then a can replace b in any mathematical expression or formula without changing its meaning. (For a formal explanation, see § In logic)
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative.
Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. [2] A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form. A quadratic form Q and its associated symmetric bilinear form B are related by the following equations:
South Korean President Yoon Suk Yeol delivers an address to the nation at the Presidential Office in Seoul, South Korea, December 7, 2024. The Presidential Office/Handout via REUTERS
The final week of the NFL regular season got off to a hot start with the Baltimore Ravens clinching the AFC North following a dominant 35-10 win over the Cleveland Browns at MT&T Bank Stadium.
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