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Unit 24Solving Quadratic Equations

Presentation 1 Factorisation

Presentation 2 Using the formula

Presentation 3 Summary of Number of Solutions

Presentation 4 Completing the Square

Unit 24Solving Quadratic Equations

- 24.1 Factorisation

A quadratic equation is of the form and can be

solved easily if it factorises.

Example 3 Solve Solution Factorising so

either or

Example 4 Solve Solution Factorising so

there is only one (repeated) solution

Example 1 Solve Solution Factorising so

either or solution or

Example 2 Solve Solution Factorising so

either or

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Unit 24Solving Quadratic Equations

- 24.2a Using the formula

There is a formula to solve the quadratic

equation The solution is given by

Solution Here so

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Example 1 Using the formula, solve the quadratic

equation Giving the solution correct to two

decimal places.

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or

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or

To 2 d.p.

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Unit 24Solving Quadratic Equations

- 24.2b Using the formula

Again, using the formula to solve the quadratic

equation Example 2 Solve the quadratic

equation Solution Here , , and

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?

?

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?

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There is only one distinct root as

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Unit 24Solving Quadratic Equations

- 24.2c Using the Formula

Again, using the formula to solve the quadratic

equation Example 3 Solve the quadratic

equation Solution Here , , and

?

?

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?

?

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This has no solution as

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This is the case when

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Unit 24Solving Quadratic Equations

- Summary of Number of Solutions

A quadratic equation can have 2,1 or 0

solutions. The following graphs illustrate these

graphically and show how the number of solutions

depend on the sign of which is part of the

quadratic formula.

No. of solutions 0 See previous Example 3

No. of solutions 1 See previous Example 2

?

No. of solutions 2 See previous Example 1

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Unit 24Solving Quadratic Equations

- 24.4 Completing the Square

The formula for solving a quadratic equation

comes from completing the square. Example Expres

s in the form when a, p and q are real

numbers. Solution

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So comparing coefficients,

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Extension What is the minimum value of ?

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This is and it occurs when

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Thus

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