New A level Maths Specifications

 
I will soon be updating my website www.mathstutormatt.com to reflect the significant changes to the A level mathematics and further mathematics curriculum.
 
New resources will cover:
 
AS Maths
A level Maths
AS Further Maths
A level Further Maths
Pure Mathematics
Core Pure Mathematics
Statistics and Mechanics
Further Statistics and Mechanics
Decision Mathematics
 
This may take some time so thank you for your patience!
 
Please get in touch if you have any specific enquiries and I will be happy to help – Matt
 
 

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Deriving The Quadratic Formula

An example of online maths tuition. The topic is deriving the quadratic formula using completing the square.

Ever wondered where the quadratic formula comes from?

Starting with the general form of a quadratic equation, you can derive the quadratic formula yourself using the technique of ‘completing the square’.

An example of online maths tuition. The topic is deriving the quadratic formula using completing the square.

As you can see in the above screenshot from a recent online maths lesson, it is possible to start with the general form of a quadratic equation ax^2 + bx + c = 0 and manipulate it into the classic quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .

The key to this derivation is the technique of completing the square. This is applied to the content of the red box in Line 2 and allows us to form the expression in Line 3.

Line 4 is all about making (x + \frac{b}{2a})^2 the subject and then creating an algebraic common denominator of 4a^2 for the expression on the right-hand side.

Line 5 involves taking the square root of both sides and then x is made the subject in the green equation in the bottom right corner.

Next, the box in the top right corner shows the square root being applied to the denominator of 4a^2 to create a new common denominator of 2a. Finally, the two algebraic fractions are combined and we’re done!

ax^2 + bx + c = 0 \rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}


Would you like help with any of the maths in this post?

Please post a comment or send me a message.


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Trigonometric functions and the unit circle

An example of online maths tuition. The topic is trigonometric functions.

The other day, a student asked me:

“When I use the trig functions, what do the numbers on my calculator really mean?”

An example of online maths tuition. The topic is trigonometric functions.

We discussed this question in detail and linked the output values from the SIN and COS calculator buttons for different angles to the horizontal and vertical measurements from the centre of the unit circle to a point P on the circumference. For example, when the angle \theta is 0 degrees, the point P has coordinates (1, 0) and then when \theta is 90 degrees, the coordinates are (0, 1).

By plotting the different values of the x and y coordinates for different points around the circumference of the circle, we were able to produce the graphs of the cosine and sine functions.

We also discussed the value of the tangent function as a ratio of the sine and cosine functions and why this leads to some of the more unsual features of the graph of tan (\theta).


Would you like help with any of the maths in this post?

Please post a comment or send me a message.


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