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IBMathsDPAnalysis & Approaches (AA)SLRevision Notes2. FunctionsQuadratic Functions & GraphsSolving Quadratics

Solving Quadratics (DP IB Analysis & Approaches (AA)): Revision Note

Dan Finlay

Author

Last updated

2 December 2024

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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 $a x^{2} + b x + c = 0$
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 $a x^{2} + b x + c = 0$
Clearly identify the values of a, b & c
Substitute the values into the formula
- $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
  - 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 $a (x - p) (x - q) = 0$
Set each factor to zero and solve
- $x - p = 0 \Rightarrow x = p$
- $x - q = 0 \Rightarrow x = q$

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

Complete the square to rewrite the quadratic equation in the form $a {(x - h)}^{2} + k = 0$
Get the squared term by itself
- ${(x - h)}^{2} = - \frac{k}{a}$
If $(- \frac{k}{a})$ is negative then there will be no solutions
If $(- \frac{k}{a})$ is positive then there will be two values for $x - h$
- $x - h = \pm \sqrt{- \frac{k}{a}}$
Solve for x
- $x = h \pm \sqrt{- \frac{k}{a}}$

Examiner Tips and Tricks

When using the quadratic formula with awkward values or fractions you may find it easier to deal with the " $b^{2} - 4 a c$ " (discriminant) first
- This can help avoid numerical and negative errors, improving accuracy

Worked Example

Solve the equations:

a) $4 x^{2} + 4 x - 15 = 0$ .

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

b) $3 x^{2} + 12 x - 5 = 0$ .

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

c) $7 - 3 x - 5 x^{2} = 0$ .

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

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Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

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.