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Flashcards · Maths

Algebra

86 cards from 5 lessons

  1. What is the difference between an expression and an equation?

    An expression has no equals sign, such as \(3x + 5\). An equation has one and can be solved, such as \(3x + 5 = 20\).

  2. What is an identity?

    An equation that is true for every value of the letter, written with \(\equiv\), such as \(2(x + 3) \equiv 2x + 6\).

  3. What are like terms?

    Terms with the same letters to the same powers. \(3x\) and \(5x\) are like terms, but \(3x\) and \(3x^2\) are not.

  4. Simplify \(4a + 3b - a + 5b\).

    \(3a + 8b\).

  5. What is \(3x \times 4x^2\)?

    \(12x^3\). Multiply the numbers and add the powers.

  6. What is \(12x^5 \div 3x^2\)?

    \(4x^3\). Divide the numbers and subtract the powers.

  7. Expand \(3(2x - 5)\).

    \(6x - 15\).

  8. Expand \(-2(x - 4)\).

    \(-2x + 8\), because a negative times a negative is a positive.

  9. What does expanding two brackets involve?

    Multiplying every term in one bracket by every term in the other, which gives four products to collect.

  10. Expand and simplify \((x + 3)(x + 5)\).

    \(x^2 + 8x + 15\).

  11. Expand and simplify \((x + 7)(x - 4)\).

    \(x^2 + 3x - 28\).

  12. Expand and simplify \((x + 3)(x - 3)\).

    \(x^2 - 9\). The middle terms cancel.

  13. What is the common mistake with \((x + 3)^2\)?

    Writing \(x^2 + 9\). It is \((x + 3)(x + 3) = x^2 + 6x + 9\), and the middle term is the one people miss.

  14. Expand and simplify \((x - 4)^2\).

    \(x^2 - 8x + 16\).

  15. How can you check an expansion?

    Put a number such as \(x = 1\) into the original and your answer. They must give the same value.

  16. How do you write an even number and an odd number using \(n\)?

    Even: \(2n\). Odd: \(2n + 1\).

  17. How do you show that the sum of two consecutive integers is odd?

    \(n + (n + 1) = 2n + 1\), which is always odd.

  18. What is the first step in any factorising question?

    Take out the highest common factor.

  19. Factorise \(6x + 15\).

    \(3(2x + 5)\).

  20. Factorise fully \(6x^2y - 9xy^2\).

    \(3xy(2x - 3y)\).

  21. How do you know an expression is fully factorised?

    Nothing is left that divides into every term inside the brackets.

  22. What two numbers do you need to factorise \(x^2 + bx + c\)?

    Two numbers that multiply to give \(c\) and add to give \(b\).

  23. Factorise \(x^2 + 7x + 12\).

    \((x + 3)(x + 4)\).

  24. Factorise \(x^2 - 2x - 15\).

    \((x - 5)(x + 3)\).

  25. How can you check a factorisation?

    Expand the brackets and see whether you get the original expression back.

  26. What is the difference of two squares rule?

    \(a^2 - b^2 = (a + b)(a - b)\).

  27. Factorise \(x^2 - 49\).

    \((x + 7)(x - 7)\).

  28. Factorise \(4x^2 - 25\).

    \((2x + 5)(2x - 5)\).

  29. Why does \(x^2 + 49\) not factorise?

    It is a sum of squares. No pair of numbers multiplies to 49 and adds to 0.

  30. How can you work out \(51^2 - 49^2\) without a calculator?

    \((51 + 49)(51 - 49) = 100 \times 2 = 200\).

  31. What is the factorised form of \(x^2 + 6x + 9\)?

    \((x + 3)^2\), a perfect square.

  32. How do you factorise \(2x^2 + 7x + 3\) (Higher tier)?

    Multiply \(2 \times 3 = 6\), split the middle term as \(6x + x\), and factorise in pairs: \((2x + 1)(x + 3)\).

  33. How do you simplify \(\dfrac{x^2 - 9}{x + 3}\) (Higher tier)?

    Factorise the top to \((x + 3)(x - 3)\), cancel the common bracket and get \(x - 3\).

  34. Why can you not cancel the \(x\) in \(\dfrac{x + 6}{x}\)?

    The \(x\) is only one term of the top, not a factor of the whole top, so the expression does not simplify to 6.

  35. What is the golden rule of solving equations?

    Do the same thing to both sides.

  36. How do you solve \(3x - 4 = 11\)?

    Add 4 to get \(3x = 15\), then divide by 3: \(x = 5\).

  37. How do you solve \(5 - 2x = 11\)?

    Subtract 5 to get \(-2x = 6\), then divide by \(-2\): \(x = -3\).

  38. What is the first step when there are unknowns on both sides?

    Collect the \(x\) terms on one side by subtracting the smaller \(x\) term from both sides.

  39. Solve \(5x + 3 = 2x + 18\).

    \(3x = 15\), so \(x = 5\).

  40. Solve \(3(2x - 1) = 4x + 9\).

    Expand to \(6x - 3 = 4x + 9\), so \(2x = 12\) and \(x = 6\).

  41. How do you solve \(\dfrac{x + 5}{3} = \dfrac{x - 1}{2}\)?

    Multiply both sides by 6 to get \(2(x + 5) = 3(x - 1)\), which gives \(x = 13\).

  42. What is the first step in forming an equation from a problem?

    Say what the letter stands for, then write the other quantities in terms of it.

  43. How do you write three consecutive integers?

    \(n\), \(n + 1\) and \(n + 2\), which add up to \(3n + 3\).

  44. What do the angles in a triangle add up to?

    \(180^\circ\), so an angle equation is set equal to 180.

  45. What do the symbols \(<\), \(>\), \(\leq\) and \(\geq\) mean?

    Less than, greater than, less than or equal to, and greater than or equal to.

  46. What is the special rule for inequalities?

    Reverse the inequality sign when you multiply or divide both sides by a negative number.

  47. Solve \(-2x > 6\).

    \(x < -3\). The sign reverses when you divide by \(-2\).

  48. What do an open circle and a filled circle mean on a number line?

    An open circle means the value is not included (\(<\) or \(>\)). A filled circle means it is included (\(\leq\) or \(\geq\)).

  49. Which integers satisfy \(-2 \leq x < 4\)?

    \(-2, -1, 0, 1, 2, 3\). The value \(-2\) is included and 4 is not.

  50. How do you write ‘at most 20’ as an inequality?

    \(x \leq 20\).

  51. How can you check the solution to an equation?

    Substitute it back into the original equation and see whether both sides match.

  52. How do you substitute a negative number safely?

    Put it in brackets. If \(a = -3\), then \(a^2 = (-3)^2 = 9\).

  53. What is the value of \(3x^2\) when \(x = -2\)?

    12. Square first (\((-2)^2 = 4\)), then multiply by 3.

  54. What is the difference between \(3x^2\) and \((3x)^2\)?

    \(3x^2 = 3 \times x^2\), but \((3x)^2 = 9x^2\).

  55. What is the formula for speed?

    Speed \(=\) distance \(\div\) time.

  56. What is the formula for the area of a trapezium?

    \(A = \tfrac{1}{2}(a + b)h\).

  57. How do you write a formula for a \(\pounds 35\) call-out plus \(\pounds 22\) per hour?

    \(C = 35 + 22h\).

  58. What does the subject of a formula mean?

    The letter on its own on one side of the equals sign, such as \(y\) in \(y = 3x + 5\).

  59. How do you rearrange a formula?

    Undo the operations in reverse order, using inverse operations and doing the same to both sides.

  60. Make \(x\) the subject of \(y = 3x + 5\).

    \(x = \dfrac{y - 5}{3}\).

  61. Make \(t\) the subject of \(v = u + at\).

    \(t = \dfrac{v - u}{a}\).

  62. Make \(x\) the subject of \(y = \dfrac{2x - 1}{3}\).

    \(x = \dfrac{3y + 1}{2}\).

  63. Make \(a\) the subject of \(P = 2(a + b)\).

    \(a = \dfrac{P}{2} - b\).

  64. Make \(m\) the subject of \(E = \tfrac{1}{2}mv^2\).

    \(m = \dfrac{2E}{v^2}\).

  65. Make \(r\) the subject of \(A = \pi r^2\) (Higher tier).

    \(r = \sqrt{\dfrac{A}{\pi}}\).

  66. What do you do when the subject appears twice (Higher tier)?

    Collect every term containing the subject on one side, factorise it out, then divide.

  67. If \(C = 12 + 8d\) and \(C = 84\), how do you find \(d\)?

    \(84 = 12 + 8d\), so \(72 = 8d\) and \(d = 9\).

  68. What is wrong with writing \(-3^2\) when you mean \((-3)^2\)?

    \(-3^2\) means \(-(3^2) = -9\). The bracket is what makes the answer 9.

  69. What is a term-to-term rule?

    A rule that tells you how to get from one term to the next, such as add 3.

  70. What is the position-to-term rule?

    The nth term: a formula that gives any term from its position \(n\), such as \(3n + 2\).

  71. What is an arithmetic sequence?

    A sequence that goes up or down by the same amount each time, called the common difference.

  72. What is a geometric sequence?

    A sequence where each term is multiplied by the same number, such as 3, 6, 12, 24.

  73. What is a Fibonacci-type sequence?

    A sequence in which each term is the sum of the two terms before it.

  74. What are the first five triangular numbers?

    1, 3, 6, 10 and 15.

  75. How do you find the nth term of a linear sequence?

    The common difference \(d\) gives \(dn\). Compare \(dn\) with the terms and adjust. For 5, 8, 11 it is \(3n + 2\).

  76. What is the nth term of 4, 7, 10, 13?

    \(3n + 1\).

  77. What is the nth term of 20, 17, 14, 11?

    \(-3n + 23\).

  78. How do you test whether a number is in a sequence?

    Set the nth term equal to the number and solve. If \(n\) is not a positive whole number, it is not in the sequence.

  79. Is 100 a term of \(3n + 2\)?

    No. \(3n = 98\) gives \(n = 32.67\ldots\), which is not a whole number.

  80. How do you find the first term of \(3n + 2\) that is over 100?

    \(3n + 2 > 100\) gives \(n > 32.67\ldots\), so \(n = 33\) and the term is 101.

  81. How do you check an nth term?

    Substitute \(n = 1\) and \(n = 2\) and see whether you get the first two terms.

  82. What is the nth term for a row of squares made from matchsticks (4, 7, 10, ...)?

    \(3n + 1\). Three sticks are added for each new square, plus one to start.

  83. How do you spot a quadratic sequence?

    The first differences change, but the second differences are constant.

  84. A quadratic sequence has second difference 2. What does its nth term start with?

    \(n^2\), because the coefficient is half the second difference (Higher tier).

  85. What is the nth term of 4, 10, 18, 28, 40 (Higher tier)?

    \(n^2 + 3n\).

  86. What is the nth term of a geometric sequence (Higher tier)?

    \(ar^{n-1}\), where \(a\) is the first term and \(r\) the common ratio. For 3, 6, 12 it is \(3 \times 2^{n-1}\).