GCSE: Quadratic Functions and Simplifying Rational Expressions

GCSE: Quadratic Functions and Simplifying Rational Expressions

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 10th December 2015

Factorising means : To turn an expression into a product of factors.

2x2 + 4xz 2x(x+2z)

x2 + 3x + 2 (x+1)(x+2)

2x3 + 3x2 – 11x – 6 (2x+1)(x-2)(x+3)

Year 8 Factorisation

Year 9 Factorisation

A Level Factorisation

Factorise

Factorise

Factorise

So what factors can we see here?

Factorising Overview

5 + 10x x – 2xz x2y – xy2 10xyz – 15x2y xyz – 2x2yz2 + x2y2

Factor Challenge

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

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Extension Question:What integer (whole number) solutions are there to the equation

Answer: . So the two expressions we’re multiplying can be This gives solutions of ?

Exercises

Factorising out an expression

It’s fine to factorise out an entire expression:

𝑥 (𝑥+2 )−3 (𝑥+2 )→(𝑥+2)(𝑥−3)

𝑥 (𝑥+1 )2+2 (𝑥+1 )→ (𝑥2+𝑥+2 ) (𝑥+1 )

2 (2𝑥−3 )2+𝑥 (2𝑥−3 )→(5 𝑥−6 )(2𝑥−3)𝑎 (2𝑐+1 )+𝑏 (2𝑐+1 )→(𝑎+𝑏)(2𝑐+1)

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Harder Factorisation

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Exercises

Edexcel GCSE Mathematics Textbook

Page 111 – Exercise 8DQ1 (right column), Q2 (right column)

Expanding two brackets

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1112

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Faster expansion of squared brackets

There’s a quick way to expand squared brackets involving two terms:

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Four different types of factorisation

1. Factoring out a term 2.

2 𝑥2+4 𝑥=2 𝑥 (𝑥+2 ) 𝑥2+4 𝑥−5=(𝑥+5 ) (𝑥−1 )

3. Difference of two squares

4 𝑥2−1=(2𝑥+1 ) (2𝑥−1 )

4.

Strategy: either split the middle term, or ‘go commando’.

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2.

Which is ?

How does this suggest we can factorise say ?

𝑥2−𝑥−30=(𝑥+5 ) (𝑥−6 )?

Is there a good strategy for working out which numbers to use?

2.

1

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3

𝝅45

67

8

910

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3. Difference of two squares

Firstly, what is the square root of:

√ 4 𝑥2=2𝑥 √25 𝑦 2=5 𝑦

√16 𝑥2𝑦 2=4 𝑥𝑦 √𝑥4 𝑦4=𝑥2 𝑦2

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√9 (𝑧−6 )2=3(𝑧−6)?

3. Difference of two squares

4 𝑥2−9

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2 𝑥2 𝑥 33√ √

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3. Difference of two squares

1−𝑥2=(1+𝑥 )(1−𝑥)

(𝑥+1 )2− (𝑥−1 )2=4 𝑥

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49− (1−𝑥 )2=(8− 𝑥)(6+𝑥)?

512−492=200?(In your head!)

18 𝑥2−50 𝑦2=2 (3 𝑥+5 𝑦 ) (3 𝑥−5 𝑦 )?

(2 𝑡+1 )2−9 (𝑡−6 )2=(5 𝑡−17 ) (−𝑡+19 )?

3. Difference of two squares

Exercises:

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4.

2 𝑥2+𝑥−3=(2 𝑥+3)(𝑥−1)Factorise using:

a. The ‘commando’ method*

b. Splitting the middle term

* Not official mathematical terminology.

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Exercises

Well Hardcore:

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1011

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‘Commando’ starts to become difficult from this question onwards.

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Simplifying Algebraic Fractions

2𝑥2+4 𝑥𝑥2−4

=2 𝑥𝑥−2

3 𝑥+3𝑥2+3 𝑥+2

=3

𝑥+2

2𝑥2−5𝑥−36 𝑥3−2𝑥4

=− 2𝑥+12𝑥3

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Negating a difference

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Exercises

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Algebraic Fractions

35 +

110=

710

23−

14=

512

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(Note: If you’ve added/subtracted fractions before using some ‘cross-multiplication’-esque method, unlearn it now, because it’s pants!)

How did we identify the new denominator to use?

Algebraic Fractions

The same principle can be applied to algebraic fractions.

1𝑥 +

2𝑥2

=𝑥𝑥2

+2𝑥2

=𝑥+2𝑥2?

1𝑥 −

2𝑥2+2𝑥

=1

𝑥+2?

The Wall of Algebraic Fraction Destiny

“To learn the secret ways of algebra ninja, simplify fraction you must.”

13𝑥+6 +

15 𝑥+10−

215 𝑥+30=

25 (𝑥+2 )

52𝑥+1

− 32 𝑥+3

=4 (𝑥+3 )

(2𝑥+1 ) (2𝑥+3 )

1𝑥+1−

1𝑥=− 1

𝑥+1?

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Recap

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Exercises

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Expand the following:

(𝑥+3 )2=𝑥2+6 𝑥+9

(𝑥+5 )2+1=𝑥2+10 𝑥+26

(𝑥−3 )2=𝑥2−6 𝑥+9

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What do you notice about the coefficient of the term in each case?

(𝑥+𝑎 )2=𝑥2+2𝑎𝑥+𝑎2?

Completing the Square – Starter

Completing the square

Typical GCSE question:“Express in the form , where and are constants.”

(𝑥+3 )2−9?

Completing the square

More examples:?

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Exercises

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Express the following in the form

11 𝑥2+2𝑎𝑥+1= (𝑥+𝑎 )2−𝑎2+1?

More complicated cases

Express the following in the form :

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Exercises

Put in the form or

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Proofs

Show that for any integer , is always even.

How many would we need to try before we’re convinced this is true? Is this a good approach?

Proofs

Prove that the sum of three consecutive integers is a multiple of 3.

We need to ensure this works for any possible 3 consecutive numbers. What could we represent the first number as to keep things generic?

Proofs

Prove that odd square numbers are always 1 more than a multiple of 4.

How would you represent…

Any odd number: 2𝑛+1

Any even number: 2𝑛

Two consecutive odd numbers. 2𝑛+1 ,2𝑛+3

One less than a multiple of 3.

3𝑛−1

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Two consecutive even numbers. 2𝑛 ,2𝑛+2?

Proofs

Prove that the difference between the squares of two odd numbers is a multiple of 8.

People in the left row work on this:

People in the middle row work on this:

People in in the right row work on this:

Example Problems

[June 2012] Prove that is a multiple of 8 for all positive integer values of .

[Nov 2012] (In the previous part of the question, you were asked to factorise , which is )

“ is a positive whole number. The expression can never be a prime number. Explain why.”

[March 2013] Prove algebraically that the difference between the squares of any two consecutive integers is equal to the sum of these two integers.

Exercises

Edexcel GCSE Mathematics Textbook

Page 469 – Exercise 28EOdd numbered questions

Even/Odd Proofs

Some proofs don’t need algebraic manipulation. They just require us to reason about when our number is odd and when our number is even.

Prove that is always odd for all integers .When is even: is . So is .

When is odd: is . So is .

Therefore is always odd.

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