Solving Quadratic Equations by Factorisation

Slideshow 19 　Mathematics

Mr Sasaki　　　 Room 307

Objectives• Understand the solutions for a quadratic

in the form • Be able to factorise quadratics to help

solve them• Be able to solve quadratic equations

where solutions may be in surd form (no co-efficient.)

Factorised QuadraticsAs you know, we can factorise quadratics to help us solve them. For simple quadratics, this is the easiest and quickest method.After we factorise something simple, we get something in the form . For the general case as shown, what are the solutions? or Why are these the solutions?

You should know that if some product , either or , right?

Of course then this means that if , either or . So or .

Other QuadraticsHow about solutions for quadratics in the form ?

How about solutions for quadratics in the form ?Once again, we can simplify this to make We would then get the two equations…

𝑏𝑥+𝑐=0or𝑚𝑥+𝑛=0

There is no change; we can divide this by and get so or .

Making the subject, we get…𝑥=−

𝑐𝑏

or𝑥=−𝑛𝑚

It’s easy to solve factorised quadratics!

PracticeSolve . or .Solve . or .Solve . or .

or or or 𝑥=8

𝑥=−1 or or or

or or

FactorisationWe know how to factorise any quadratic in the form where .If , we can solve for after factorising.

Example

Factorise and hence, solve (𝑥+2 ) (𝑥−7 )=0 or .

Example

Factorise and hence, solve 𝑎 ∙𝑏=−30 ,𝑎+𝑏=7⇒𝑎=,𝑏=¿10 −3

𝑥=32,𝑥=−5

Answers - Easy

or or or or

𝑥=±2𝑥=±7 𝑥=−7 𝑥=4or or or or

or or or or

Answers - Hardor or or or

or or 𝑥=±

12𝑥=±

34

or or or or

𝑥=−12𝑥=

43 𝑥=−

32

or

FactorisationWe know how to solve equations in the form .Example

Solve .

2 𝑥2=12⇒𝑥2=6⇒𝑥=√6 so no need to write . (Remember in Japanese maths, .

Everyone can do this. Can we write the expression in the form

No coefficientExample

Write in the form where , . Do not simplify.

2 𝑥2=12⇒2 𝑥2−12=0⇒ 2(𝑥¿¿2−6)=0¿⇒ 2(𝑥+√6 )(𝑥−√6)=0

As there is no coefficient, we can use the principle that

Note: Never simplify! You should always write as the coefficient. So for the above question the answer is not .

No coefficientLet’s try one more example.Example

Write in the form where , . Do not simplify.

3 𝑥2=60⇒3 𝑥2−60=0⇒ 3(𝑥¿¿2−20)=0¿⇒ 3(𝑥+√20)(𝑥−√20)=0⇒ 3(𝑥+2√5)(𝑥−2√5)=0

Answers - Top

2 (𝑥−√7 ) (𝑥+√7 )=0𝑥=(±)√7

3 (𝑥−2√2 ) (𝑥+2√2 )=0𝑥=2√2

4 (𝑥−√2 ) (𝑥+√2 )=0𝑥=√2

3 (𝑥−3√5 ) (𝑥+3√5 )=0𝑥=3√5

2(𝑥− √22 )(𝑥+ √2

2 )=0

𝑥=√22

24 (𝑥− √24 )(𝑥+ √2

4 )=0

𝑥=√24

Answers - Bottom

2(𝑥− √66 )(𝑥+ √6

6 )=0𝑥=√6

6

3 (𝑥− √612 )(𝑥+ √6

12 )=0𝑥=√6

12

5(𝑥− √360 )(𝑥+ √3

60 )=0𝑥=√3

60

2(𝑥− √54 )(𝑥+ √5

4 )=0𝑥=√5

4The solution for would be imaginary.