EDEXCEL GCSE MATHEMATICS

Finding the LCM of two numbers when you have the prime factors

• List all the prime factors out

• If a factor appears more than once, list it that many times, e.g. 2, 2, 2, 3, 4 and 2, 2, 3, 4 would be 2, 2, 2, 3, 4

• Multiply these together to get the LCM

Finding the HCF of two numbers when you have the prime factors

• List all the prime factors that appear in both numbers

• Multiply these together

Multiplying fractions

Multiply the top and bottom separately

Dividing fractions

Turn the second fraction upside down then multiply

Rule for terminating and recurring decimals

If the denominator has prime factors of only 2 or 5, it is a terminal decimal

Turning a recurring decimal into a fraction

• Name the decimal with an algebraic letter

• Multiply by a power of ten to get the one loop of repeated numbers past the decimal point

• Subtract the larger value from the single value to get an integer

• Rearrange

• Simplify

Turning a recurring fraction into a decimal when the recurring decimal is not immediately after the decimal, e.g. r = 0.16666...

• Name the decimal with an algebraic letter e.g. r = 0.16666...

• Multiply by a power of ten to get the non-repeating part out of the bracket e.g. 10r = 1.6666...

• Multiply to get the repeating part out of the bracket e.g. 100r = 16.6666...

• Take away the larger value from the smaller one (to get an integer) e.g. 100r - 10r = 90r = 15

r = 15/90

• Simplify e.g. 15/90 = 1/6

Turning a fraction into a decimal

• Make the fraction have all nines at the bottom

• The number on the top is the recurring part, the number of nines is the number of recurring decimals there are

Significant figures

The first number which isn't a zero. This is rounded.

Rules for calculating with significant digits

Estimating square roots

• Find two numbers either side of the number in the root

• Make a sensible estimate depending on which one it is closer to

Truncated units

When a measurement is truncated, the actual measurement can be up to a whole unit bigger but no smaller, e.g. 2.4 truncated to 1 d.p. is 2.4 ≤ x < 2.5

Multiplying and dividing standard form

• Convert both numbers to standard form

• Separate the power of ten and the other number

• Do each calculation separately

Adding and subtracting standard form

• Convert both numbers into standard form

• Make both powers of 10 the same in each bracket

• Add the two numbers and multiply by whatever power of ten; they are to the same power so this can be done

Negative powers

1 over whatever the number to the power was, e.g.

7⁻² = 1 / 7² = 1 / 49

a⁻⁴ = 1 / a⁴

If the number is a fraction, then it is swapped around, e.g.

(3/5)⁻² = (5/3)² = 25 / 9

Fractional powers

Something to the power of 1/2 means square root

Something to the power of 1/3 means cube root

Something to the power of 1/4 means fourth root, e.g.

25^½ = √25 = 5

Two-stage fractional powers

When there is a fraction with a numerator higher than one, spilt it into a fraction and a power and do the root first, then power, e.g.

64^5/6 = (64^1/6)⁵ = (2)⁵ = 32

Difference between two squares

a²-b²=(a+b)(a-b)

Simplifying surds

Split the number in the root into a square number and the lowest other number possible, e.g.

√250 = √(25 × 10) = 5√10

Rationalising the denominator

This is done to get rid of a surd on the denominator. You multiply by the same fraction of the surd, but with the operation the other way round.

Removing fractions when they (the fractions) appear on both sides of an equation

• Multiply by the lowest common multiple of both numbers

• Simplify

Quadratic formula

Completing the square

• Write out in the form ax²+bx+c

• Write out the first bracket in the form

(x + b/2)²

• Multiply out the brackets and add or subtract to make the number outside the bracket match the original

Completing the square when 'a' isn't one

• Factorise with the 'a' value outside the brackets

• Write out in the form a(x+b/2)

• Add or subtract the remaining number

Facts about completing the square

For a positive quadratic,

• The number outside the bracket is the y value of a graph

• The number inside the bracket is the x value multiplied by -1

Adding and subtracting algebraic fractions

• Make both denominators equal by multiplying the numerators by the other denominator

• Simplify

Finding the nth term of quadratic sequence

• Find the difference between the difference in each term (this can be called a second difference)

• Divide the second difference by two to get the coefficient of the n² term

• Take away the n² term from the original sequence to get a linear sequence, which can be easily worked out

Algebra with inequalities exception

Whenever you multiply or divide by a negative number, flip the inequality sign round

Inequalities on number lines

• Open circles for < or >

• Closed circles for ≤ or ≥

Quadratic inequalities general rule

• If x² > a² then x > a or x < -a

• If x² < a² then -a < x < a

(this is because with a square root the number can be positive or negative, and the negative would mean flipping the sign)

Drawing quadratic inequalities

• Factorise the equation when it equals zero to find the x intercepts

• Sketch the graph, paying attention to if the x² coefficient is positive or negative

• Solve the equation using the graph, paying attention to whether the original equation is a < or > (or a ≥ or ≤)

• If the equation asks for x > 0, then solve it for where the quadratic is above the x-axis. On a negative graph, this will give the results as -a < x < b, and on a positive graph it will result in x < -a and x > b

• If the equation asks for x < 0, then solve it for when the quadratic is below the x-axis. On a negative graph, the results will be x < -a or x > b, and on a positive graph it would be -a < x < b

• In the picture, the equation is x² - x - 2, as the x intercepts are -1 and 2. The answer to x² - x - 2 > 0 would be x < -1 and x > 2

• If the inequality has a ≤ or ≥ then the final answer has to include a ≤ or ≥ for both roots

Showing an inequality on a graph

• Convert the inequality into an equation

• Draw the graph for the equation

• Work out which side of the line you want and shade the region which it satisfies (it usually asks in the question)

• If the inequality has a < or >, then draw a dotted line

• If the inequality has a ≤ or ≥, then draw a solid line

Composite functions (e.g. fg(x))

• Always do the function close to x first, e.g. fg(x) would be f(g(x))

Inverse functions (written as f⁻¹(x))

• Write out the equation as x = f(y)