Equations, formulae and identities

Common error: A power of a power means multiply the indices (3 × 4 = 12), not add them. The number 2 must also be raised to the power 4, not multiplied by 4.

Exam tip: "Completely" means the bracket must have no common factor left. Check the terms inside: 2x and 3y share nothing, so you are done.

Common error: (a − b)² is not a² − b². Write the bracket out twice and multiply every term, so you get both middle terms.

Common error: Squaring a negative number gives a positive result: (−3)² = 9, not −9. Put negative values in brackets when you substitute so the signs stay visible.

Common error: A formula must start with the subject and an equals sign: write C = 12 + 3.5h, not just 12 + 3.5h, and not C = 15.5h.

Exam tip: Check your point in BOTH equations: 3(3) − 4 = 5 and 2(3) + 5 = 11. A quick check catches swapped coordinates.

Common error: After factorising to (x − 7)(x + 4) = 0, the solutions have the opposite signs to the numbers in the brackets: x = 7 and x = −4. Substitute back to check.

Common error: When you multiply or divide both sides of an inequality by a negative number, the inequality sign must be reversed. Check by substituting a value, e.g. x = −5 gives 7 + 15 = 22 > 19.

Exam tip: On a number line, a closed circle means the end value is included (≤ or ≥) and an open circle means it is not (< or >). Do the same operation to all three parts of a double inequality.

Exam tip: In part (b) you can divide both sides by 3 first. This avoids expanding the bracket and gives fewer chances to slip.

Common error: In part (b), do not divide only the \(2w\) by 3. Multiply the whole of both sides by 3 to clear the fraction first.

Exam tip: After finding \(x\), substitute it into all three expressions. Then check that they add to 180° before you choose the largest.

Exam tip: The \(y\) terms have opposite signs, so add the equations. Substitute back into the simpler equation to find \(y\).

Common error: Multiply both terms inside the bracket by 4. Writing \(8x - 3\) instead of \(8x - 12\) loses the accuracy marks.

Common error: If (x + 3)(x + 4) = 0, the solutions are x = −3 and x = −4, not +3 and +4. Set each bracket equal to zero and solve it.

Exam tip: Check which end of the inequality is included. Here −2 is included because of ≤, but 4 is not included because of <.

Common error: Dividing both sides of x² = 6x by x loses the solution x = 0. Factorise instead. Taking a square root also gives a negative solution as well as a positive one.

Exam tip: Undo the operations in reverse order: remove the added term first, then divide. Doing the division first leaves you with u/a, which is wrong.

Common error: Writing h = 2A/a + b loses the accuracy mark. Keep (a + b) together in brackets as the denominator.

Common error: Fewer machines means more time. Using direct proportion here (15 × 6 ÷ 8 = 11.25 hours) gives an answer that should be spotted as unreasonable.

Exam tip: After factorising, expand your answer in your head to check that you get back the original expression.

Common error: For (p⁵)², multiply the powers to get p¹⁰; adding them to give p⁷ is wrong.

Exam tip: For x² + bx + c, look for two numbers that multiply to c and add to b: here −4 and −5 multiply to 20 and add to −9.

Common error: −2 × −4 = +8. Writing −8 when expanding −2(x − 4) is the most frequent slip in this type of question.

Common error: Working out (4.5 + 9.8) × 3 = 42.9 ignores the order of operations: the multiplication at is done first.

Exam tip: Finding the cost of one item first (the unitary method) works for every part of a direct proportion question.

Exam tip: When a quadratic models a length, state clearly which solution you reject and why. Then use the valid value to answer the question that was actually asked.

Exam tip: Always write the final formula with k replaced by its value, e.g. y = 3.5x, not just k = 3.5.

Common error: When subtracting a whole expression, put it in brackets: −(x² + 3x − 10) gives +10, not −10.

Exam tip: Check a factorisation by expanding it again: the brackets must give back every term, including the signs.

Common error: A minus sign in front of a fraction applies to the whole numerator: −3(x − 2) = −3x + 6. Also multiply the right-hand side by the common denominator.

Exam tip: Collect the number terms carefully, keeping each sign with its number: +10 − 20 + 40 − 10 = +20. Substituting your answer back to check the angle sum takes only a few seconds.

Exam tip: When you rearrange, undo the operations in reverse order: subtract 32 first, then multiply by 5, then divide by 9. Test your rearranged formula with the pair you found in part (a).

Common error: Both people get 12 years older. Ana's future age is x + 12, so 'twice as old' means 2(x + 12), not 2x + 12.

Exam tip: In a 'show that' part, write every step of the expansion and simplification; the final line must match the given expression exactly.

Common error: Keep the units consistent: either work entirely in pence (1500 + 8m) or entirely in pounds (15 + 0.08m). Mixing £15 with 8m pence gives a wrong equation.

Exam tip: Re-read what the question actually asks for. Here you need a + c, not just one of the prices.

Common error: Dividing the bill by the number of GB (27 ÷ 6) ignores the fixed fee. Two unknowns need two equations.

Exam tip: Check your answer in the context: 21 + 10 = 31 baskets and 2(21) + 3(10) = 72 points.

Exam tip: When a quadratic comes from a length, always say why you reject the negative root, then answer the question actually asked (here the perimeter, not just x).

Common error: Consecutive odd numbers differ by 2, not 1, so the second number is n + 2, not n + 1.

Common error: Check each end separately: < means the end value is excluded and ≤ means it is included. Remember that 0 and negative integers count too.

Common error: t > 300 does not include 300: at exactly 300 texts both plans cost £21. The least whole number satisfying a strict inequality is the next integer.

Exam tip: In a "show that" part, every step must be written down, including the collected expression for the perimeter, so that the given result clearly follows.

Exam tip: Check your answers in the equation you did not use to find the second value. Here, \(2(8.50) + 5(5.50) = 44.50\) confirms both values.

Exam tip: Expand the bracket after clearing the fraction. This puts the \(m\) term on its own and makes the rearrangement much easier.

Exam tip: When one equation already gives \(y\) in terms of \(x\), substitution is quicker than elimination.

Common error: Add 12 to both people's ages. Writing \(3x + 12 = 2x + 12\) forgets that the 2 multiplies Priya's future age, \(x + 12\).

Common error: Forgetting the final square root leaves v² = 90, giving a 'speed' of 90 m/s. The square root must cover the whole fraction.

Common error: (n + 3)² is not n² + 9. Write it as (n + 3)(n + 3) and expand all four terms.

Exam tip: Divide every term by 5 before factorising. This gives the simpler equation t² − 4t + 3 = 0, which is much easier to factorise.

Common error: When you multiply \(\frac{x-1}{3}\) by 12, the result is \(4(x - 1) = 4x - 4\), not \(4x - 1\). Keep the numerator in a bracket.

Exam tip: Undo the operations in reverse order: clear the fraction first by multiplying by 5, then deal with the −2, then divide by 3.

Common error: After making the coefficients of \(y\) equal, look at their signs. Here they are \(+12y\) and \(-12y\), so you must add the equations, not subtract. Check your answers by substituting into the other equation: \(2(3) - 4(-2.5) = 16\).

Common error: The minus sign in front of the second fraction multiplies the whole numerator. So \(-2(2x + 5) = -4x - 10\), not \(-4x + 10\).

Exam tip: Subtract the two equations in part (a) to eliminate \(c\) straight away: \(9 = 3m\).

Common error: (2x + 1)² is not 4x² + 1. Write it as (2x + 1)(2x + 1) and expand fully to get 4x² + 4x + 1.

Common error: Writing x > ±7 is wrong. The inequality x² > 49 has two separate regions: x 7.

Exam tip: Deal with a fractional negative index in three steps: the denominator is a root, the numerator is a power, and the minus sign means take the reciprocal.

Exam tip: When the unknown is in the index, rewrite both sides as powers of the same base, then equate the indices.

Common error: Isolate v² completely before taking the square root, and take the root of the whole right-hand side: √(2E/m), not √(2E)/m.

Common error: Using simple direct proportion (scaling 18 by 50/30) ignores the square. Always write the equation with the constant k, find k, then substitute.

Exam tip: When the unknown is under a square root, find √T first and then square it as a separate, clearly written step.

Exam tip: Write the general equation with k first and show the substitution. That line earns credit even if a later calculation slips.

Common error: Multiplying by d instead of d² when finding k: for inverse square, k = F × d² = 720 × 25.

Exam tip: Write down a, b and c with their signs before substituting, and put brackets round negative values: −4 × 3 × (−4) = +48.

Common error: When completing the square, (x + 4)² = x² + 8x + 16, so you must subtract 16 to keep the expression the same.

Exam tip: "Hence" means use your completed-square form: (x − a)² = b gives x = a ± √b in one step. The same form gives the turning point: y = (x − a)² − b has a minimum at (a, −b), so P = b − (x − a)² has a maximum at (a, b).

Exam tip: After finding the critical values, sketch the parabola (or test a value such as x = 0) to decide whether the solution lies between the roots or outside them.

Exam tip: A proof needs a general argument: factorise to show the factor you need and state the conclusion in words. Testing a few numbers is not a proof.

Exam tip: Sketch the U-shaped curve mentally. For '≥ 0' you need the parts of the curve on or above the x-axis. These are the two outer regions, not the region between the roots.

Common error: The turning point of (x − 4)² − 13 is at x = +4, not x = −4. The x-coordinate is the value that makes the bracket equal to zero.

Common error: Every term must be multiplied by the common denominator, including the 1 on the right-hand side. That side becomes (x − 1)(x + 2), not 1.

Common error: Using d = kt instead of d = kt² gives k = 9.8 and every later answer is wrong. Read 'square of' carefully.

Exam tip: Check the answer makes sense: the intensity has fallen to one ninth, so the distance should have trebled (5 m to 15 m).

Exam tip: For part (c), use the proportion directly: multiplying L by 4 multiplies √L by √4 = 2. No new constant is needed.

Common error: The reciprocal of 1/f − 1/u is NOT f − u. Combine into one fraction first, then turn it upside down.

Exam tip: Split the middle term: find two numbers that multiply to 2 × −15 = −30 and add to 7 (10 and −3), then factorise in pairs.

Exam tip: Expand two brackets first and simplify before multiplying by the third; write all six terms out before collecting like terms.

Common error: Only whole factors can be cancelled. Crossing out the x² terms from top and bottom is not valid.

Exam tip: When the new subject appears twice, gather every term containing it on one side, then take it out as a common factor.

Exam tip: When the subject appears twice, gather every term containing it on one side and factorise it out. Check by substituting back: (3×11 + 2)/(11 − 4) = 35/7 = 5.

Exam tip: Substitute the x-values into the linear equation to find the y-values. It is quicker than using the quadratic and less likely to cause errors.

Common error: With c = −4, the term −4ac is −4 × 3 × (−4) = +48. This gives 25 + 48 = 73, not 25 − 48. Put negative values in brackets when you substitute.

Exam tip: Only whole factors can be cancelled. Factorise the top and the bottom completely first, then look for an identical bracket in both.

Common error: When the x² coefficient is not 1, take it out of the x terms first and remember to multiply the −9 by that factor (2 × 9 = 18) before combining with 23.

Common error: When subtracting fractions, the minus sign applies to the whole second numerator: −2(x + 2) = −2x − 4, not −2x + 4.

Exam tip: When the new subject appears twice, follow four steps: clear the fraction, expand, collect every term containing the subject on one side, then factorise it out.

Exam tip: For a 'what happens if' part, apply the proportion rule to the multiplier: h × 4 gives D × √4 = D × 2.

Exam tip: 'Hence' means use the result you have just found. Spot which expressions play the roles of x and y instead of starting again.

Exam tip: Time = distance ÷ speed. Write each time as a fraction first, form the equation in words (longer time − shorter time = 1), then multiply every term by the common denominator.

Common error: Expanding (x + 2)² as x² + 4 loses the middle term 4x. Write (x + 2)(x + 2) and expand fully.

Exam tip: To show a line and a curve do not meet, form the quadratic and show it has no real solutions: a square equal to a negative number, or a negative value under the square root in the formula.

Exam tip: Keep unrounded values in your calculator for the subtraction in the last step. Rounding each time first (1.99 − 0.462 = 1.528) can make the final answer wrong.

Exam tip: Solving a quadratic inequality gives a mathematical answer; in a context question, always check whether part of it is impossible (here a negative speed) and reject that part with a reason.

Common error: Points on a boundary line count only for ≤ or ≥. Here y > 1 and x + y < 8 are strict, so points on those lines, such as (2, 1) and (2, 6), are not allowed.

Common error: Using n and n + 1 for two even numbers does not work: start by writing a general even number as 2n and an odd number as 2n + 1.

Exam tip: Put both solution sets on a number line. The integers you need lie where the two regions overlap. Remember that 5 is excluded because the inequality is strict.

Common error: A 20% increase in x does not give a 60% increase in y. Multipliers are cubed, not percentages multiplied by 3.

Exam tip: Use a different letter for each constant of proportionality so the two constants are not confused when you combine the formulae.

Exam tip: Factorise every numerator and denominator completely before cancelling; a difference of two squares often hides in expressions like 4x² − 1.

Exam tip: Remove the square root first by squaring, then clear the fraction. Check your answer by substituting back: with m = 5 and n = 4, √(9/1) = 3 as required.

Common error: Expand \((x + 1)^2\) as \(x^2 + 2x + 1\), not \(x^2 + 1\). Missing the middle term gives the wrong quadratic.

Exam tip: In an identity, compare coefficients of each power of x separately: the x² and constant terms give a and b directly, then the x term gives c. When asked to factorise fully, check for a common factor in every bracket.

Common error: When taking out the factor 2, remember the −9 inside the bracket also gets doubled: 2(x − 3)² means subtracting 18, not 9.