What is Roots of equations: Definition and 12 Discussions

In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities. They form a set of n points in the complex plane. This article concerns the geometry of these points, that is the information about their localization in the complex plane that can be deduced from the degree and the coefficients of the polynomial.
Some of these geometrical properties are related to a single polynomial, such as upper bounds on the absolute values of the roots, which define a disk containing all roots, or lower bounds on the distance between two roots. Such bounds are widely used for root-finding algorithms for polynomials, either for tuning them, or for computing their computational complexity
Some other properties are probabilistic, such as the expected number of real roots of a random polynomial of degree n with real coefficients, which is less than

1
+

2
π

ln

(
n
)

{\displaystyle 1+{\frac {2}{\pi }}\ln(n)}
for n sufficiently large.

p
(
x
)
=

a

0

+

a

1

x
+

+

a

n

x

n

,

{\displaystyle p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n},}
where

a

0

,

,

a

n

{\displaystyle a_{0},\dots ,a_{n}}
are real or complex numbers and

a

n

0

{\displaystyle a_{n}\neq 0}
; thus n is the degree of the polynomial.

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1. I Find the roots of the quadratic equation by differentiation

The Solution of the Quadratic Equation By Differentiation Method
2. Quadratic equation and its roots

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3. To prove that a given quadratic has integral roots

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4. Three () distinct real roots of a quadratic equation

It is given that ##x_1, x_2\; \text{and}\; x_3## are roots of the equation ##ax^2+bx+c=0##, which are pairwise distinct. If indeed they are roots, we should have ##ax_1^2+bx_1+c= 0 = ax_2^2+bx_2+c= 0 = ax_3^2+bx_3+c= 0##. On subtracting the first two, we obtain ##a(x_1^2-x_2^2)+b(x_1-x_2) =...
5. I What method should I use to get the roots of this equation?

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6. I Where does the exponential function come from in roots?

For example, in linear differential equations, there might be these questions where we'd directly use e∫pdx as the integrating factor and then substitute it in this really cliche formula but I never really understood where it came from. Help ?
7. Got stuck due to the inequality not being satisfied

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8. Symmetries of graphs and roots of equations

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9. MHB Roots of Equations & Sum of Inverses: $a=1,2,3,\dots,2011$

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10. Why do we care so much about the roots of equations?

Hi, I know what roots are and how to find them but I don’t know why they are so important. What is that makes the points where a function become zero so important? I saw a similar post on this topic, but it talks about roots from an optimization point of view. However, finding roots...
11. Finding Multiple Roots of Equations

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12. Finding Multiple Roots of Equations

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