- 3a means 3 times a; a b means a times b; a squared means a times a.
- Write terms without multiplication signs, and put the number before the letter (3a, not a3).
- a over b means a divided by b; a term is a part of an expression separated by plus or minus.
- An expression has no equals sign; an equation does.
Watch out: Writing "less than" or "more than" statements in the wrong order, such as writing 6 − y instead of y − 6 for "6 less than y".
- Like terms have exactly the same letters and powers, such as 3x and 5x.
- Add or subtract only the coefficients; the letter part stays the same.
- 3x and 3x squared are not like terms and cannot be combined.
- Watch the signs: the sign belongs to the term that follows it.
Watch out: Combining terms that are not like terms, such as adding 4a and 3b together as if they were the same type of term.
- Multiply every term inside the bracket by the term outside.
- A minus sign outside changes the sign of every term inside.
- x(x + 3) gives x squared plus 3x, not x squared plus 3.
- After expanding, collect any like terms.
Watch out: Forgetting to multiply every term inside the bracket by the number outside, especially the second term.
- Multiply each term in the first bracket by each term in the second (FOIL, or a grid).
- (x + 3)(x + 5) gives x squared plus 8x plus 15.
- The x term comes from adding the two products; the number term from multiplying them.
- (x + a) squared means (x + a)(x + a), which is x squared plus 2ax plus a squared, not x squared plus a squared.
Watch out: Forgetting one of the four multiplications when expanding, most often the "outer" or "inner" pair of terms.
- Factorising is the reverse of expanding: you put the brackets back in.
- Take out the highest common factor of all the terms.
- Include the letters: the common factor of 6x squared and 9x is 3x.
- Check by expanding your answer; it should give the original expression.
Watch out: Not finding the highest common factor, and instead factoring out a smaller factor, which leaves the expression not fully factorised.
- For x squared plus b x plus c, find two numbers that multiply to c and add to b.
- Those two numbers go straight into the brackets: (x + p)(x + q).
- If c is positive both numbers share the sign of b; if c is negative they have opposite signs.
- When the x squared coefficient is not 1, split the middle term or use the grid method.
Watch out: Finding two numbers that multiply to the coefficient of x instead of the constant term, or add to the constant term instead of the coefficient of x.
- a squared minus b squared always factorises to (a + b)(a - b).
- It only works for a subtraction of two square terms with no middle term.
- x squared minus 9 is (x + 3)(x - 3); 4x squared minus 25 is (2x + 5)(2x - 5).
- Spotting this is often much faster than the standard method.
Watch out: Trying to apply the difference of two squares to an addition, such as x² + 16, when the pattern only applies to a subtraction of two squares.
- Write x squared plus b x plus c as (x + b over 2) squared minus (b over 2) squared plus c.
- The form (x + p) squared plus q gives the turning point at (minus p, q).
- It is one way to solve a quadratic that will not factorise nicely.
- If the coefficient of x squared is not 1, factor it out first.
Watch out: Getting the sign inside the bracket wrong, such as writing (x + 4)² for x² − 8x + ... instead of (x − 4)².
- Do the same operation to both sides to keep the equation balanced.
- Undo operations in reverse order: deal with plus and minus before times and divide.
- For letters on both sides, collect the letters on one side and the numbers on the other.
- Check your answer by substituting it back into the original equation.
Watch out: Doing an operation to only one side of the equation, instead of applying it to both sides to keep the equation balanced.
- Rearrange the equation into the form a x squared plus b x plus c equals 0 first.
- If it factorises, set each bracket equal to 0 to get the two solutions.
- A quadratic usually has two solutions, sometimes one repeated, sometimes none real.
- Never divide both sides by x; you would lose the solution x = 0.
Watch out: When solving x² = a, only giving the positive square root and forgetting that the negative square root is also a solution.
- x equals minus b plus or minus the root of (b squared minus 4 a c), all over 2a.
- It solves any quadratic written as a x squared plus b x plus c equals 0.
- The plus or minus gives the two solutions.
- The discriminant b squared minus 4 a c tells you the number of real roots: positive means two, zero means one, negative means none.
Watch out: Substituting a value with the wrong sign into the formula, especially when b or c is negative.
- Simultaneous equations are solved together; the answer is the pair of values that fits both.
- Elimination: make the coefficients of one letter match, then add or subtract to remove it.
- Substitution: rearrange one equation for a letter and put it into the other.
- A linear and a quadratic pair usually has two solution pairs.
Watch out: Only multiplying part of an equation when scaling it for elimination, forgetting to multiply the number on the right-hand side as well.
- Solve an inequality like an equation, but keep the inequality sign.
- If you multiply or divide both sides by a negative number, flip the sign.
- Show the solution on a number line: an open circle for a strict inequality, a filled circle for "or equal to".
- Integer solutions are the whole numbers that satisfy the inequality.
Watch out: Using a closed circle for a strict inequality (< or >) instead of an open circle, or the reverse for ≤ or ≥.
- Solve the matching quadratic equation first to find the critical values.
- Sketch the parabola to see where it is above or below the x-axis.
- "Less than 0" means between the roots; "greater than 0" means outside the roots.
- Give the answer as an inequality or in set notation, not just the critical values.
Watch out: Solving the quadratic to find the critical values but not checking which side of them actually satisfies the original inequality, instead guessing based on the inequality symbol alone.
- Changing the subject means getting one letter on its own, using the same steps as solving an equation.
- Undo operations in reverse order and do the same to both sides.
- If the new subject appears twice, collect those terms and factorise.
- To remove a square root, square both sides; to remove a square, take the root.
Watch out: Not applying the same operation to every term on both sides of the formula, especially when a side has more than one term.
- Factorise the top and bottom first, then cancel common brackets.
- To add or subtract, use a common denominator, usually the product of the two denominators.
- To divide, flip the second fraction and multiply.
- You can only cancel whole factors, never individual terms.
Watch out: Cancelling a term from a fraction without factorising first, such as cancelling an x that is added rather than multiplied throughout the numerator or denominator.
- f(x) is a rule; f(3) means substitute x = 3 into that rule.
- f g(x) means do g first, then apply f to the result.
- The inverse undoes the function: swap x and y, then rearrange for y.
- The domain is the set of allowed inputs; the range is the set of possible outputs.
Watch out: Making an arithmetic slip with negative numbers when substituting a value into a function, especially when the input itself is negative.
- A term-to-term rule tells you how to get from one term to the next.
- For a linear sequence, the nth term is the common difference times n, plus a number.
- Find that number by working backwards from the first term.
- Substitute a term number into the nth term rule to test a value or find a later term.
Watch out: Miscounting the common difference, especially in a decreasing sequence, forgetting to include the negative sign.
- A quadratic sequence has a constant second difference, and the n squared coefficient is half of it.
- Find the n squared part first, subtract it from the sequence, then find the linear rule for what is left.
- A geometric sequence multiplies by a fixed common ratio each time.
- The nth term of a geometric sequence is a times r to the power (n minus 1).
Watch out: Making an arithmetic slip when evaluating a quadratic nth term, especially forgetting to square n before multiplying or adding the other terms.
- Use 2n for any even number and 2n plus 1 for any odd number.
- Consecutive integers are n, n plus 1, n plus 2.
- Show the general case with algebra; giving a few examples is not a proof.
- Finish by explaining why your expression must be even, odd, or a multiple.
Watch out: Using specific numbers to "prove" a statement, which only shows it works for those examples, not that it is true for every possible value.
