IBMaths: AA HLDPRevision Notes2. Functions2.2 Quadratic Functions & Graphs2.2.3 Solving Quadratics

Solving Quadratics (DP IB Maths: AA HL)

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Solving Quadratic Equations

How do I decide the best method to solve a quadratic equation?

  • A quadratic equation is of the form a x squared plus b x plus c equals 0
  • If it is a calculator paper then use your GDC to solve the quadratic
  • If it is a non-calculator paper then...
    • you can always use the quadratic formula
    • you can factorise if it can be factorised with integers
    • you can always complete the square

How do I solve a quadratic equation by the quadratic formula?

  • If necessary rewrite in the form a x squared plus b x plus c equals 0
  • Clearly identify the values of a, b & c
  • Substitute the values into the formula
    • 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 
      • This is given in the formula booklet
  • Simplify the solutions as much as possible

How do I solve a quadratic equation by factorising?

  • Factorise to rewrite the quadratic equation in the form a left parenthesis x minus p right parenthesis left parenthesis x minus q right parenthesis equals 0
  • Set each factor to zero and solve
    • x minus p equals 0 rightwards double arrow x equals p
    • x minus q equals 0 rightwards double arrow x equals q

How do I solve a quadratic equation by completing the square?

  • Complete the square to rewrite the quadratic equation in the form a open parentheses x minus h close parentheses squared plus k equals 0
  • Get the squared term by itself
    • open parentheses x minus h close parentheses squared equals negative k over a
  • If open parentheses negative k over a close parentheses is negative then there will be no solutions
  • If open parentheses negative k over a close parentheses is positive then there will be two values for x minus h
    • x minus h equals plus-or-minus square root of negative k over a end root
  • Solve for x
    • x equals h plus-or-minus square root of negative k over a end root

Examiner Tip

  • When using the quadratic formula with awkward values or fractions you may find it easier to deal with the " b squared minus 4 a c " (discriminant) first
    • This can help avoid numerical and negative errors, improving accuracy

Worked example

Solve the equations:

a)
4 x squared plus 4 x minus 15 equals 0.

2-2-3-ib-aa-sl-quadratic-equations-a-we-solution

b)
3 x squared plus 12 x minus 5 equals 0.

2-2-3-ib-aa-sl-quadratic-equations-b-we-solution

c)
7 minus 3 x minus 5 x squared equals 0.

2-2-3-ib-aa-sl-quadratic-equations-c-we-solution

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    Dan

    Author: Dan

    Expertise: Maths

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.