2.1 Algebraic manipulation
Algebraic manipulation
Everything on algebraic manipulation that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise algebraic manipulation free
A worked example
Collecting like terms · [2 marks]
Simplify this expression: 6m + n + 8n + 7m
- B1Collects the m terms: 13m
- B1Collects the n terms, giving 13m + 9n
The exam skills (19)
Collecting like terms
- Like terms have exactly the same letters.
- Collect each letter separately, keeping the sign in front of each term.
- Write the answer with no like terms left.
Watch out: Losing a minus sign: the sign belongs to the term after it.
Multiplying terms
- Multiply the numbers.
- Multiply the letters: add the powers of the same letter.
- Write numbers first, then letters in order.
Watch out: Writing cd × cd = 2cd. It is c²d².
Expanding and simplifying
- Multiply everything inside each bracket by the number outside.
- Watch the signs: − × − = +.
- Collect like terms.
Watch out: Only multiplying the first term in the bracket.
Factorising
- Find the highest common factor of the numbers.
- Write it outside the bracket.
- Divide each term by it to fill the bracket. Expand to check.
Watch out: Taking out a factor that is not the highest. "Fully" means the HCF.
Substitution
- Replace each letter with its number, in brackets.
- Use BIDMAS.
- Be careful with negatives: 6 × (−6) = −36.
Watch out: Writing 2c with c = −6 as 2 − 6. It means 2 × (−6).
Writing an expression
- Turn each phrase into maths: "twice as many" is 2x, "3 more" is + 3.
- Build it up in the order the sentence says.
- Check with a number for x.
Watch out: Writing 1 + 2 × x in the wrong order. Test it with x = 5.
Double brackets
- Multiply each term in the first bracket by each term in the second (FOIL or a grid).
- That gives four terms.
- Collect the two x terms.
Watch out: Only multiplying first × first and last × last.
Factorising a quadratic
- Find two numbers that multiply to the last number and add to the middle one.
- Write (x + p)(x + q).
- Expand to check.
Watch out: Getting the signs wrong. Check by expanding.
Difference of two squares
- a² − b² = (a + b)(a − b).
- Square root each part.
- One bracket with +, one with −.
Watch out: Trying to find two numbers that add to a middle term that is not there. Spot the difference of two squares.
Changing the subject
- Undo what is done to the letter, in reverse order.
- Do the same thing to both sides each time.
- End with the letter on its own on one side.
Watch out: Doing the steps in the wrong order: undo + and − before × and ÷.
Multiplying algebraic fractions
- Multiply the tops and multiply the bottoms.
- Cancel numbers and letters that are on the top and bottom.
- Use the index laws for powers.
Watch out: Cancelling terms that are added. You can only cancel factors.
Algebraic proof
- Write the integers with algebra: n, n + 1, n + 2.
- Add them and simplify.
- Factorise to show the multiple, and say why it proves it.
Watch out: Trying numbers. Examples are not a proof.
Three brackets
- Expand two of the brackets first.
- Multiply your answer by the third bracket.
- Collect like terms, in order of powers.
Watch out: Making an error in the first expansion and not checking it.
Harder quadratic
- Multiply a × c. Find two numbers that multiply to that and add to b.
- Split the middle term using those numbers.
- Factorise in pairs, then take out the common bracket.
Watch out: Using (x + p)(x + q) when there is a number in front of x².
Algebraic fraction
- Factorise the top.
- Factorise the bottom.
- Cancel the common bracket.
Watch out: Cancelling the x² terms. Only whole brackets cancel.
Adding algebraic fractions
- Common denominator: multiply the bottoms.
- Multiply each top by the other bottom.
- Add the tops and simplify.
Watch out: Adding the tops and the bottoms.
Subject appears twice
- Multiply out the fraction so nothing is on the bottom.
- Get every term with the letter on one side, everything else on the other.
- Factorise the letter out, then divide.
Watch out: Trying to divide before the letter appears only once.
Completing the square
- Halve the x number: x² − 14x starts as (x − 7)².
- Take away that number squared: − 49.
- Add on the constant from the question.
Watch out: Adding the squared number instead of subtracting it.
Show that
- Start with the left-hand side only.
- Expand and simplify, showing every line.
- Finish with exactly the right-hand side.
Watch out: Working on both sides at once. "Show that" means one side becomes the other.
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