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Flashcards · Maths

Further Algebra

80 cards from 5 lessons

  1. What fact makes solving by factorising work?

    If two things multiply to give zero, one of them must be zero.

  2. What is the first step when solving a quadratic by factorising?

    Rearrange so that one side is zero.

  3. Solve \((x - 2)(x - 3) = 0\).

    \(x = 2\) or \(x = 3\).

  4. Solve \(x^2 - 49 = 0\).

    \(x = 7\) or \(x = -7\).

  5. Solve \(x^2 - 6x = 0\).

    \(x = 0\) or \(x = 6\).

  6. Why should you not divide both sides of \(x^2 = 6x\) by \(x\)?

    You lose the solution \(x = 0\).

  7. Factorise \(x^2 - 5x + 6\).

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

  8. How do you factorise \(2x^2 + 7x + 3\)?

    Multiply \(a\) and \(c\) to get 6, split the middle term into \(6x + x\), and factorise in pairs to get \((2x + 1)(x + 3)\).

  9. How are the solutions of a quadratic linked to its graph?

    They are the \(x\)-values where the graph crosses the \(x\)-axis.

  10. What is the quadratic formula (Higher tier)?

    \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

  11. What is the discriminant (Higher tier)?

    \(b^2 - 4ac\), the number under the square root.

  12. What does a negative discriminant mean (Higher tier)?

    There are no real solutions.

  13. Complete the square for \(x^2 + 6x\) (Higher tier).

    \((x + 3)^2 - 9\).

  14. Why might you reject one solution of a quadratic?

    It may not make sense in the problem, such as a negative length.

  15. How do you check the solutions?

    Substitute each one back into the original equation.

  16. What does a repeated root look like on a graph?

    The curve just touches the \(x\)-axis.

  17. What are simultaneous equations?

    Two or more equations that must be true at the same time.

  18. What does the point where two graphs cross represent?

    The solution of the simultaneous equations.

  19. When do you subtract the equations in elimination?

    When the terms in one letter are the same in both.

  20. When do you add the equations in elimination?

    When the terms in one letter are opposites.

  21. What do you do if no letter has matching numbers?

    Multiply one or both equations to make a pair match.

  22. What is the first step in the substitution method?

    Get one letter on its own in one equation.

  23. Solve \(x + y = 9\) and \(x - y = 1\).

    \(x = 5\), \(y = 4\).

  24. How should you check your solution?

    Substitute both values into the equation you did not use.

  25. What does it mean if two lines are parallel?

    The equations have no solution.

  26. How do you write a word problem as simultaneous equations?

    Define the letters and write one equation for each statement.

  27. What do you do first with a line and a quadratic (Higher tier)?

    Substitute the line into the quadratic to get a quadratic in \(x\).

  28. How many solutions can a line and a circle have (Higher tier)?

    Two, one or none.

  29. Solve \(y = x^2\) and \(y = x + 6\) (Higher tier).

    \((3, 9)\) and \((-2, 4)\).

  30. Why use brackets when substituting an expression?

    They stop sign errors.

  31. How many pairs of values solve a pair of linear equations?

    One, unless the lines are parallel or the same.

  32. What is the equation of the line through \((0, 1)\) with gradient 1?

    \(y = x + 1\).

  33. What does a solid line mean on a graph of an inequality?

    The line is included, so the inequality is \(\leq\) or \(\geq\).

  34. What does a dashed line mean?

    The line is not included, so the inequality is \(<\) or \(>\).

  35. How do you decide which side of a line to shade?

    Test a point, such as the origin, in the inequality.

  36. What does \(x \geq 1\) look like on a graph?

    The region to the right of the vertical line \(x = 1\), including the line.

  37. What does \(y \leq 2x\) look like on a graph?

    The region on or below the line \(y = 2x\).

  38. How do you describe a region bounded by three lines?

    Write one inequality for each boundary.

  39. What are the integers satisfying \(-2 < x \leq 3\)?

    \(-1, 0, 1, 2, 3\).

  40. Do points on a dashed line count as inside the region?

    No.

  41. How many integer points satisfy \(x \geq 1\), \(y \geq 1\), \(x + y \leq 6\)?

    15.

  42. What does the origin test show for \(x + y \leq 6\)?

    \(0 + 0 \leq 6\) is true, so shade the origin side.

  43. How do you solve a quadratic inequality (Higher tier)?

    Find the roots and use a sketch of the graph.

  44. Solve \(x^2 - x - 6 < 0\) (Higher tier).

    \(-2 < x < 3\).

  45. Solve \(x^2 > 9\) (Higher tier).

    \(x < -3\) or \(x > 3\).

  46. What does R usually stand for in these questions?

    The region that satisfies all the inequalities.

  47. Where does the line \(x + y = 6\) cross the axes?

    \((6, 0)\) and \((0, 6)\).

  48. Why check with a point inside the region?

    All the inequalities must be true there.

  49. What is a surd?

    A root that cannot be written as a whole number or fraction, such as \(\sqrt{2}\).

  50. Why are surds left as roots?

    Their decimals never end, so the root is the exact value.

  51. What is \(\sqrt{a} \times \sqrt{a}\)?

    \(a\).

  52. What is the product rule for surds?

    \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\).

  53. Simplify \(\sqrt{12}\).

    \(2\sqrt{3}\).

  54. Simplify \(\sqrt{50}\).

    \(5\sqrt{2}\).

  55. Simplify \(\sqrt{72}\).

    \(6\sqrt{2}\).

  56. Which square numbers are useful for simplifying surds?

    4, 9, 16, 25, 36 and 49.

  57. What is \(3\sqrt{2} + 5\sqrt{2}\)?

    \(8\sqrt{2}\).

  58. Can \(\sqrt{2} + \sqrt{3}\) be simplified?

    No, because they are not like surds.

  59. What is \(\sqrt{2} \times \sqrt{8}\)?

    4.

  60. What does rationalising the denominator mean?

    Rewriting a fraction so that there is no surd on the bottom.

  61. Rationalise \(\dfrac{6}{\sqrt{3}}\).

    \(2\sqrt{3}\).

  62. What is the conjugate of \(2 + \sqrt{3}\)?

    \(2 - \sqrt{3}\).

  63. What is \((2 + \sqrt{3})(2 - \sqrt{3})\)?

    1.

  64. Expand \((2 + \sqrt{5})^2\).

    \(9 + 4\sqrt{5}\).

  65. What may you cancel in an algebraic fraction?

    Factors, never terms that are added or subtracted.

  66. What is the first step in simplifying an algebraic fraction?

    Factorise the top and the bottom.

  67. Simplify \(\dfrac{x^2 - 9}{x + 3}\) (Higher tier).

    \(x - 3\).

  68. Is \(\dfrac{x + 3}{3}\) equal to \(x\)?

    No, the 3 on top is added, so it cannot be cancelled.

  69. How do you add \(\dfrac{1}{x} + \dfrac{1}{x + 1}\) (Higher tier)?

    Use the common denominator \(x(x + 1)\) to get \(\dfrac{2x + 1}{x(x + 1)}\).

  70. How do you divide algebraic fractions (Higher tier)?

    Turn the second fraction upside down and multiply.

  71. How do you solve an equation with fractions?

    Multiply every term by the common denominator.

  72. Solve \(\dfrac{x - 1}{3} + \dfrac{x + 2}{6} = 2\).

    \(x = 4\).

  73. How do you write an even number with \(n\)?

    \(2n\).

  74. How do you write an odd number with \(n\)?

    \(2n + 1\).

  75. How do you write three consecutive numbers?

    \(n\), \(n + 1\), \(n + 2\).

  76. What does \(\equiv\) mean?

    The expressions are equal for every value of the letters, an identity.

  77. Is testing a few numbers a proof?

    No, you must use algebra to show it is true for all numbers.

  78. How do you disprove a statement?

    Find one counter-example.

  79. Give a counter-example to "\(n^2 + n + 1\) is always prime".

    \(n = 4\) gives 21, which is \(3 \times 7\).

  80. How should a proof end?

    With a sentence stating what has been shown.