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If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative. On the other hand, r 1 – r 2 and r 1 – r 3 are complex conjugates, and their product is real and positive. [ 23 ]
In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.
The product (a + bi) · (c + di) can be calculated in the following way. k 1 = c · (a + b) k 2 = a · (d − c) k 3 = b · (c + d) Real part = k 1 − k 3 Imaginary part = k 1 + k 2. This algorithm uses only three multiplications, rather than four, and five additions or subtractions rather than two.
In the simple case of a function of one variable, say, h(x), we can solve an equation of the form h(x) = c for some constant c by considering what is known as the inverse function of h. Given a function h : A → B, the inverse function, denoted h −1 and defined as h −1 : B → A, is a function such that
How to Solve It (1945) is a small volume by mathematician George Pólya, describing methods of problem solving. [1] This book has remained in print continually since ...
Then we can substitute again, letting x = b and y = c, to show that if bc = 0 then b = 0 or c = 0. Therefore, if abc = 0, then a = 0 or (b = 0 or c = 0), so abc = 0 implies a = 0 or b = 0 or c = 0. If the original fact were stated as "ab = 0 implies a = 0 or b = 0", then when saying "consider abc = 0," we would have a conflict of terms when ...
You may have heard that Social Security is facing financial trouble because Congress and presidents raided the trust funds and wondered how such a thing could be allowed to happen.
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.