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Show that x^2 - x +2 is positive for all values of x

(x-1/2) 2 = (x-1/2)(x-1/2)=x 2 -x+1/4, add 7/4 to get x 2 -x+2 therefore x 2 -x+2 = (x-1/2) 2 + 7/4 squared numbers always positive so (x-1/2) 2 >= 0 therefore (x-1/2) 2 + 7/4 is always positive
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Answered by Pearse M. • Maths tutor
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Vectors a and b are defined by a = 2i + 3j and b = 4i - 2j, find 3a-b in terms of i and j

3a = 6i + 9j -b = -4i + 2j 3a-b = (6i-4i) + (9j+2j) = 2i + 11j
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Answered by Pearse M. • Maths tutor
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Find ∫(4x - 2/(x^3)) dx

Using the integer rule 1/ (x n ) = x -n question becomes ∫(4x - 2(x -3 )) dx Then using the formula ∫x n dx = (x n+1 )/n+1 and assuming there may be an integer value (+c) the answer is 2(x 2 ) + x -2 + c
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Answered by Pearse M. • Maths tutor
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How to write an algebraic fraction in a given form e.g. (3+13x-6x^2)/(2x-3) as Ax + B + C/(2x-3) where A, B and C are natural numbers

I would break this down into simple steps: Step 1 - Equate the 2 expressions: (3 + 13x - 6x 2 ) / (2x - 3) = Ax + B + C / (2x - 3) Step 2 - Multiply both sides by the demoninator: 3 + 13x - 6x 2 = (2x - 3)*A...
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Answered by Alex T. • Maths tutor
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Find the derivative of the function f:(0,oo)->R, f(x)=x^x.

The domain of the function allows us to write f(x)=x x as f(x)=e ln(x ^x) =e x ln( x) (since ln(x) is defined on (0,oo) only). Using the standard derivative rules we get f'(x)=e x ln(x) (x ln(x))'=e x ln(x) ...
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Answered by Andrei R. • Maths tutor
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