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GCSEMathsAQAHigherFlashcards2. AlgebraQuadratic Equations

Quadratic Equations (AQA GCSE Maths)

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  • What is a quadratic equation?

    A quadratic equation is an equation of the form a x squared plus b x plus c equals 0,
    where a, b, and c are constants, and a is not equal to zero.

  • What does it mean to solve a quadratic equation by factorising?

    Solving a quadratic equation by factorising means:

    1. Making sure it is in the form a x squared plus b x plus c equals 0 (i.e. with zero on one side).

    2. Factorising the quadratic.

    3. Setting each bracket equal to zero to find the solutions.

  • True or False?

    If  open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses equals 0,  then either  x plus 4 equals 0  or  x minus 1 equals 0.

    True.

    If  open parentheses x plus 4 close parentheses open parentheses x minus 1 close parentheses equals 0,  then either  x plus 4 equals 0  or  x minus 1 equals 0.

  • True or false?

    The solutions of the equation open parentheses x minus 2 close parentheses open parentheses x plus 5 close parentheses equals 0 are x equals negative 2 and x equals 5.

    False.

    To find the solutions of the equation open parentheses x minus 2 close parentheses open parentheses x plus 5 close parentheses equals 0, you should solve the linear equations x minus 2 equals 0 and x plus 5 equals 0.

    The solutions are x equals 2 and x equals negative 5.

    Note that the signs in front of 2 and 5 are the 'other way round' to how they appear in the brackets.

  • What two linear equations should you solve to find the solutions of the quadratic equation open parentheses 8 x plus 7 close parentheses open parentheses 2 x minus 3 close parentheses equals 0?

    To find the solutions of the quadratic equation  open parentheses 8 x plus 7 close parentheses open parentheses 2 x minus 3 close parentheses equals 0,  you should solve the linear equations  8 x plus 7 equals 0  and  2 x minus 3 equals 0.

    The solutions are x equals negative 7 over 8 and x equals 3 over 2.

  • True or false?

    x equals 0 is one of the solutions of the quadratic equation x open parentheses 5 x minus 1 close parentheses equals 0.

    True.

    If x open parentheses 5 x minus 1 close parentheses equals 0, then either x equals 0 or 5 x minus 1 equals 0.

    So x equals 0 is one of the solutions.

    The other solution is x equals 1 fifth.

  • What is the quadratic formula?

    The quadratic formula can be used to solve a quadratic equation of the form  a x squared plus b x plus c equals 0.

    The quadratic formula gives the solutions as   x equals fraction numerator negative b plus-or-minus square root of b squared minus 4 a c end root over denominator 2 a end fraction.

    The quadratic formula is on the exam formula sheet, so you don't need to memorise it. But you do need to be able to use it.

  • True or False?

    You need 'equals 0' on one side of a quadratic equation to use the quadratic formula.

    True.

    You need 'equals 0' on one side of a quadratic equation to use the quadratic formula.

  • What does it mean if b squared minus 4 a c is negative when using the quadratic formula?

    If b squared minus 4 a c is negative when using the quadratic formula, it means there are no real solutions to the quadratic equation.

  • If an equation is given in completed-square form, such as open parentheses x plus 3 close parentheses squared minus 4 equals 0, explain how to solve it.

    If an equation is given in completed-square form, such as open parentheses x plus 3 close parentheses squared minus 4 equals 0, to solve it you need to make x the subject.

    Add 4 to both sides: open parentheses x plus 3 close parentheses squared equals 4

    Take plus-or-minus square roots: x plus 3 equals plus-or-minus 2

    Make x the subject: x equals negative 3 plus-or-minus 2 giving x equals negative 1 or x equals negative 5

    (Note that to solve it, you do not expand the brackets back out!)

  • When should you solve a quadratic equation by factorisation?

    You should solve a quadratic equation by factorisation when:

    • the question asks you to solve by factorisation,

    • solving two-term quadratic equations,

    • a calculator gives solutions as whole numbers or fractions (indicating that the quadratic factorises).

  • When should you use the quadratic formula to solve a quadratic equation?

    You should use the quadratic formula to solve a quadratic equation when:

    • the question asks to give solutions correct to a given accuracy, e.g. 2 decimal places,

    • you think the quadratic formula might be faster than factorising,

    • in doubt (because the quadratic formula always works!)

  • When should you solve a quadratic equation by completing the square?

    You should solve a quadratic equation by completing the square, when:

    • one part of a question says to complete the square and another part says to use the first part to solve the equation

    • making x the subject of harder formulae containing x squared and
      x terms

    (Note that, like the quadratic formula, completing the square will always work to solve a quadratic equation. It's just not always easy to do!)

  • True or False?

    Calculators can be used to check final solutions when solving quadratic equations.

    True.

    Calculators can be used to check final solutions when solving quadratic equations.

    Make sure that you know how to use your particular calculator model to solve quadratic equations.

  • What does it tell you if a calculator gives solutions to a quadratic equation as whole numbers or fractions?

    If a calculator gives solutions to a quadratic equation as whole numbers or fractions (with no square roots), it tells you that the quadratic equation can be factorised.

  • True or False?

    When solving a quadratic equation of the form x open parentheses a x plus b close parentheses equals 0, the first step is to divide both sides by x.

    False.

    It is a common mistake to divide both sides by x at the beginning when solving a quadratic equation of the form x open parentheses a x plus b close parentheses equals 0

    It is a mistake because if you do this you will lose the x equals 0 solution.

    Instead, solve x equals 0 and a x plus b equals 0.

  • True or False?

    When solving a quadratic equation of the form a x squared plus b x plus c equals 0 by completing the square, the first step should be to divide both sides by a.

    True.

    When solving a quadratic equation of the form a x squared plus b x plus c equals 0 by completing the square, you should divide both sides by a first.

  • True or False?

    The quadratic formula actually comes from completing the square to solve a x squared plus b x plus c equals 0.

    True.

    The quadratic formula actually comes from completing the square to solve a x squared plus b x plus c equals 0.

    (You don't need to be able to do this on the exam. But doing this is a great way to practise your algebra skills!)

Previous:Linear EquationsNext:Simultaneous Equations

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation