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y = 2x^2 + 4x - 8y = 2( x^2 + 2x - 4)Completing the square:y = 2( (x+1)^2 - 1 - 4 )y = 2(x+1)^2 -10the vertex of graph lies at the minimum value of y, and this occurs when x = -1:y = 2(-1+1)^2 -10y = -10t...

So we have the equation initially in the form 3^(5x-2)=4^(6-x), and as the solution involves log10, then  a sensible first move would be to take log10 of both sides, giving log10(3^(5x-2)) = log10(4^(6...

Starting on the left hand side we can expand out the brackets to get:
(sinx + cosx)(sinx + cosx)
sin²x+sinxcosx+sinxcosx+cos²x
Grouping together the like terms we can ...

Tricky. Definitely can't do it by inspection (we don't know any fuction that just differentaties to ln(x)), it's not like we've really got anything to substitute u for if we wanted to do it by substitu...

The first thing we notice is that this differential equation is seperable, meaning we can get all of our y's on the left with a dy and all of our x's on the right with a dx. Doing this by multiplying b...