Basic algebra
Posted in Maths, Core 1This topic is a recap of what you should already know from your GCSE or equivalent qualification. You are expected to know and use these techniques in multiple questions in the exam without explicitly being told to do so.
Making the subject of equations
You should be able to rearrange equations to make one letter or term the subject. The key to this is to make sure what you do on side you repeat on the other.
Example question: Make y the subject of:
Worked answer:
Start by multiplying both sides by 8 to remove the fraction on the right hand side:
Then divide both sides by 2:
Finally square root both sides to get y on its own:
Multiplying out brackets
To successfully multiply out brackets you need to make sure that each term is multiplied by every other term in the expression.
Example question: Expand and simplify: 3(x2 + 4x + 9)
Worked answer: With only one bracket you need to multiply each term in the bracket by the term on the outside, in this case 3:
3(x2 + 4x + 9) = 3 X x2 + 3 X 4x + 3 X 9 = 3x2 + 12x + 27
Example question: Expand and simplify: (x + 4)(x - 3)
Worked answer: With the two brackets this time you need to be a bit more careful. Each term in the first bracket much be multiplied by each term in the second.
(x + 4)(x - 3) = x X x + x X -3 + 4 X x + 4 X -3 = x2 + -3x + 4x + -12 = x2 + x - 12
Factoring quadratics
Factorising is the opposite of multiplying out, it requires you to rewrite an expression using brackets. A quadratic expression is written in the form ax2 + bx + c and typically factorises into two brackets.
Example question: Factorise x2 + 3x - 10
Worked answer: To start you need to create an x2 this is created from an x in both brackets:
(x )(x )
You then have a bit of trial and error: You need to find two numbers that multiply to give -10 and add to give +3, of course remembering to also think of negative numbers.
In this example 5 and -2 multiply to give -10 and add to give +3, therefore:
(x + 5)(x - 2)
The difference of two squares
A special quadratic is known as the different of two squares which states that a quadratic in the form a2 - b2 factorises to give (a + b)(a - b). For example x2 - 49 = x2 - 72 so factorises to (x + 7)(x - 7)