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ABCD is a rectangle with sides of lengths x centimetres and (x − 2) centimetres.If the area of ABCD is less than 15 cm^2 , determine the range of possible values of x.

First you interpret the given information and create an equation based on the question. x(x-2)<15. Then you express that equation in standard quadratic form: x^2-2x-15<0. Then you have to not forget th...
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Prove that 8 times any triangle number is always 1 less than a square number

A triangle number is a number such that it is the sum of n consecutive integers, starting from 0. Eg 1, 1+2, 1+2+3... are the first 3 triangle numbers. The formula for the nth triangle number is well-known a...
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Co-ordinate Geometry A-level: The equation of a circle is x^2+y^2+6x-2y-10=0, find the centre and radius of the circle, the co-ordinates of point(s) where y=2x-3 meets the circle and hence state what we can deduce about the relationship between them.

We have that x 2 + y 2 + 6x - 2y - 10 = (x+3) 2 + (y-1) 2 - 20 = 0 (step was motivated by the equation of a general circle (x-a) 2 + (y-b) 2 = r 2 )We know that the general circle has centre at (a,b) and the...
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Given the circumference x^2 - 2x + y^2 = 3, find the position of the center P and the value of the Radius. Then find the intercepts with the y axis and the tangent to the circumference at the positive y intercept.

Find center and radius From completing the quadratic: x^2 - 2x + 1 - 1 + y^2 = 3(x-1)^2 + y^2 = 4 hence, center P(1, 0) and radius R = sqrt(4) = 2 Find y intercept Intercept with y axis, then x = 0hence1 + y...
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Find y if dy/dx = y² sec²(x), given that y(0) = 1

1/y² dy/dx = sec²(x)∫ 1/y² dy/dx dx = ∫ sec²(x) dx-1/y + C 1 = tan(x) + C 2 y = -1/(tan(x) + A) where A = C 2 - C 1 y(0) = -1/A so y(0) = 1 means A = -1. Finished!
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