COMPLEX NUMBERS AND QUADRATIC EQUATIONS100 MATHEMATICS
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The problem of choice is simplified considerably by using $y=x+x^{-1}$ which satisfies a quadratic equation and therefore has only two values. For our desired value of $x$ the expression $y>0$ and hence the positive root $y$ is chosen. And in reality we are interested in the value...
1 Why do we need Complex Numbers? Some equations have no real solutions. For instance, the quadratic equation: $ - has no real solution because there is no real number that can be squared to produce . Why do we need Complex Numbers? To overcome this deficiency, mathematicians created an ...
Complex numbers provide solutions to many math, science, and engineering problems that would otherwise have no solutions. For instance, consider finding the roots of the quadratic equation: y = x2 + 4x + 1. When graphed in the x-y plane, one readily sees that the graph never intersects ...
Quadratic equations have the form ax2+ bx+ c. We can solve them by applying the quadratic formula to obtain their roots. The roots can be complex if the determinant of the quadratic formula is less than zero. The quadratic formula uses the coefficients of the terms of the equation, whose...
History ComplexNumbers Realsolution Complexsolution Figure1.2 Geometric representation quadraticequation way we can think complexnumber plane.11 1732Leonhard Euler calculated visualisethemas planarpolygon. Further breakthroughs came Abrahamde Moivre (1730) againEuler (1748), who introduced famousformulas (cos ...
Solutions to Quadratic Equations We know that often solutions to our quadratic equations result in answers such as −5 , which isn’t a real number. We can write it as an imaginary number. Principal Square Root If a is a positive number, then the principal square root of the negative nu...
wasn’t able to solve it. This was solved by an Italian mathematician called Gerolamo Cardano who found the negative roots of cubic and quadratic polynomial expressions using Complex Numbers. Complex Numbers have many uses in scientific research, fluid dynamics, quantum mechanics and signal processing...
For instance, consider finding the roots of the quadratic equation: y = x2 + 4x + 1. When graphed in the x-y plane, one readily sees that the graph never intersects the x-axis and thus has no real roots. With the use of complex numbers, however, this equation can be shown to ...