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Expand (1+x)^3. Express (1+i)^3 in the form a+bi. Hence, or otherwise, verify that x = 1+i satisfies the equation: x^3+2*x-4i = 0.

First, we factor out one (x+1). (1+x)^3 = (x+1)^2(x+1)= Then, we expand using the formula (a+b)^2 = a^2 + 2ab +b^2: =(x^2+2x+1)(x+1)= Then we multiply: = x^3 + 2x^2 + x + x^2 + 2x + 1 We sum the terms wit...

Answered by Anastasia I. • Further Mathematics tutor

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Finding modulus and argument of complex number (x+iy)

Draw argand diagram

Modulus=|z|= length of the line

So |z|=(x²+y²)^0.5

Argument=Angle between real (x axis) and line

so arg(z)=arctan(y/x)

Answered by Emma F. • Further Mathematics tutor

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a) Find the general solution to the differential equation: f(x)=y''-12y'-13y=8. b) Given that when x=0, y=0 and y'=1, find the particular solution to f(x).

First consider y''-12y'-13y=0.

Try: y=e^mx. This gives y'=me^mxand y''=m²e^mx.

Substituting into the differntial equation we get: m²e

Answered by Antonia P. • Further Mathematics tutor

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How do I prove that the differential of coshx is equal to sinhx?

Before this proof, it is important to appreciate that both of these hyperbolic functions can be written in terms of e^x. Therefore, before you begin to differentiate, you must represent coshx as (e^x + e...

Answered by Thomasina R. • Further Mathematics tutor

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