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Firstly we need to expand out y into a series of terms to make it easier for us to compute the derivative . You multiply out the brackets to get y=(x³-3x²+4x-12)/2x then we divide ea...

z* is the complex conjugate of z therefore z* = x - yi. So 16 + 32i = 5(x + yi)-3(x - yi), real: 16 = 5x - 3x => 16=2x => x=8, imaginary: 32 = 5y + 3y => 32 = 8y => y=4, therefore z = 8 + 4i.<...

f(x)=(12x^2)(e^2x) - 14, so using the chain rule f'(x)=(24x)(e^2x) + (12x^2)(2e^2x).To find the turning points set f'(x)=0, so (24x)(e^2x) + (24x^2)(e^2x) = 0. Thus (24xe^2x)(1+x)=0. Thus x=0 or x=-1.

To rationalise a surd means to manipulate a fraction involving terms which are being square rooted so that no term in the denominator is being square rooted. For this example 2/root(x) we multiply the fra...

(x + 2)²+3