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A curve has the equation y = 4x^2 + 5x + 3 and a line has the equation y = x + 2. Show that the line and the curve have one point of intersection.

Set the equations equal to each other: 4x^2 + 5x + 3 = x + 2Collect terms and set equal to 0: 4x^2 + 4x + 1 = 0Factorise the equation: (2x + 1)(2x + 1) = 0Can now find the value of x: 2x + 1 = 0, therefor...

Answered by Francesca G. • Maths tutor

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A curve has the equation y=x^2+4x+4 and a line has the equation y=2x+3. Show the line and curve have only one point of intersection and find its coordinate..

First set the equations equal to each other: x^2+4x+4 = 2x+3.Rearrange for x in form ax^2+bx+c : x^2+2x+1=0Factorise: (x+1)^2=0. Repeated root, hence only one intersection. x=-1. Using y=2x+3, y=1. So coo...

Answered by Ewan F. • Maths tutor

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Calculate (7 + √3)(7 - √3)

(7 + √3)(7 - √3) = 7² + 7√3 - 7√3 - (√3)² = 49 - 3 = 46Alternatively you can go straight to identifying that this is the difference between two squares, using the fact that (a + b)(a...

Answered by Isabel B. • Maths tutor

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Show that the lines y=3x+7 and 2y–6x=8 are parallel. Do not use a graphical method.

y=3x+7 2y-6x=8
2y-6x=82y=8+6xy= 4+3xy=3x+4When both equations equal y they both have a 3 in front of their x meaning that they both have a gradient of 3.

Answered by Jordan M. • Maths tutor

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A hemisphere is placed on top of an upside down cone. The cone has height 9cm and the hemisphere has radius 3cm. The total volume of this composite solid is x cm^3. Calculate the value of x, leaving your answer in terms of π.

To work out the total volume of the composite solid, we need the volumes of both the cone and the hemisphere. GCSE Maths students are expected to know these respective formulae; volume of a sphere = ⁴/₃πr...

Answered by Elliot C. • Maths tutor

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