Unit 25Solving Equations

Presentation 1 Algebraic Fractions

Presentation 2 Algebraic Fractions and Quadratic Equations

Presentation 3 Solving Simultaneous Equations:One Linear and One Quadratic

Presentation 4 Note

Unit 25Solving Equations

25.1 Algebraic Fractions

You can sometimes simplify fractions when the numerator and denominator are factorial.

Example

Simplify

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Unit 25Solving Equations

25.2 Algebraic Fractions and Quadratic Equations

Solution

First combine the two fractions on the LHS by putting them over a common denominator.

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Simplify to give

Divide each term by 4

Factorise

Either or

So or

ExampleSolve

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Multiply both sides by to give

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Unit 25Solving Equations

25.3 Solving Simultaneous Equations:One Linear and One Quadratic

To solve simultaneous equations when one is linear and one quadratic, you have to first eliminate one variable. You can do this by substituting one of the variables from the linear equation into the quadratic equation, forming a new quadratic in just one variable.

Example

Solve the simultaneous equation This is a quadratic that can be factorised to give

Solution

You can eliminate y by taking one equation from the other to give

or

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So or

and or ?

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Either or etc.

Unit 25Solving Equations

25.4 Note

NoteThe solution , or , are

in fact the points of intersection of the line

with the quadratic function

This is shown in the graph below