How many solutions over the complex number system
De moivre’s theorem example | find 3 distinct solutions
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and I is a symbol known as the imaginary unit, and which fulfills the equation i2 = -1 in mathematics. René Descartes nicknamed me an imaginary number because no “real” number can satisfy this equation. The real part of the complex number a + bi is called the real part, and the imaginary part is called the imaginary part. The set of complex numbers can be represented by one of the symbols or C. Despite their historical label of “imaginary,” complex numbers are regarded as just as “absolute” as real numbers in the mathematical sciences, and are essential in many areas of scientific explanation of the natural world.  [a] [a]
All polynomial equations, including those with no real-number solutions, can be solved using complex numbers. The fundamental theorem of algebra states that every polynomial equation with real or complex coefficients has a complex number as its solution. For instance, consider the equation
Complex number solutions to polynomial equations
(3x+5)/(4x+2)=(3x+4)/(4x+7)=(3x+5)/(4x+2)=(3x+4)/(4x+7) The equation x(7)+14x(5)+16x(3)+30x-560=0 has how many actual solutions? do you have? (a) a. (2) a (3) three (4) 5 The equation 5 tan (-1)x+3cot(-1)x=2 pi has how many solutions? Determine if each of the quadratic equations below has two distinct real roots. Justicy’s response: …… Consider the following x and y equation: (2)+(4x+3y-4)(2)+(x-2y-1)(4x+3y-4)=0 (x-2y-1)(4x+3y-4)=0 (x-2y-1)(4x+3y-4)=0 (x-2y-1)(4x+3y-4)=0 (x-2 What a man… Solve the problem and check your answer:
22. The equations method 0 = 4x+6y + 8z 3x + 2y+ z = 0 7x + 8y + 9z = 0 7x + 8y + 9z = 0 has (a) no solution (b) on… How many natural numbers with a maximum of four digits have a maximum of five digits in their sum? …… The equation x(7)+14x(5)+16x(3)+30x-560=0 has how many actual solutions? determine the polynomial’s value When x=3 and even when x=-3, the formula is 3x(3)-4x(2)+7x-5. For a given complex, how many geometrical isomers are possible? Pt(gly)Cl (2)BrI] Pt(gly)Cl (2)BrI] Pt(gly)C (-) [If you’re a] The polynomials 3×4-7×3+x2-x+9,4-2×2/3+3×4/5 are biquadratic polynomials. x polynomial (5) +2 times (2) +3x+7x(4)+x(3)+1 +3x+7x(4)+x(3)+1 The following is the list in ascending order: The system of equations ||z + 4 |-|z-3i|| =5 has how many solutions? and has |z| = 4? Check if the first polynomial is a by dividing the second polynomial by the first polynomial. Check if the first polynomial is a by dividing the second polynomial by the first polynomial. Determine if each of the quadratic equations below has two distinct real roots. Justicy’s response: …… How many positive integers x exist where 3X has three digits and 4X has four digits? 4×4-3×3+6×2, 4×3+4x-3, 4×3+4x-3, 4×3+4x-3, 4×3+4x-3, 4×3+4x-3, 4×3+4x-3
Complex solutions of z^4-4z^2+16=0
The following questions will help us focus on the content in this section as we review it. We should be able to write precise, coherent answers to these questions after learning this section since we should understand the principles that these questions are focused on.
We will find solutions to the quadratic equation (ax2+ bx + c = 0) using the quadratic formula (x = dfrac-b pm sqrtb2 – 4ac2a). The solutions to the equation (x2 + x + 1 = 0), for example, are
This solution has an immediate flaw since there is no real number (t) with the property that (t2 = -3), or (t = sqrt-3). We use complex numbers to make sense of solutions like this. While complex numbers naturally occur when solving quadratic equations, the problem of solving cubic equations was the catalyst for their introduction into mathematics.
We get the solutions (x = dfrac-1 + sqrt-32) and (x = dfrac-1 – sqrt-32) respectively. These are complex numbers, and we have a specific way of writing them down. We write them in such a way that the square root of is isolated (-1\). As an example, the number
Solve equations with complex numbers
The square root of negative one is involved in complex numbers, and most non-mathematicians find it difficult to believe that such a number is important. Real numbers, on the other hand, they believe, have a clear and intuitive sense. What is the best way to demonstrate to a non-mathematician that complex numbers, including real numbers, are important and meaningful?
This is not a Platonic concern about the existence of mathematics or whether abstractions are as real as tangible beings, but rather an effort to close a gap in understanding that many people have when they first encounter complex numbers. Although provocative, the wording was chosen to represent the way many people actually ask this question.
“Wait, what about $x2 = -1$?” we finally inquired. Since this was the only question left, we wanted to make up “imaginary” numbers to answer it. All of the other numbers didn’t exist and didn’t seem “true” at one point, but they’re now perfect. We can solve any polynomial now that we have imaginary numbers, so it makes sense that this is the last place to end.