Cambridge Additional Mathematics

70 Quadratics (Chapter 3)

While the Indians had knowledge of the quadratic formula even at this

early stage, it took somewhat longer for the quadratic formula to arrive in

Europe.

Around 820 AD, the Islamic mathematicianMuhammad bin Musa

Al-Khwarizmi, who was familiar with the work of Brahmagupta,

recognised that for a quadratic equation to have real solutions, the value

b^2 ¡ 4 ac could not be negative. Al-Khwarizmi’s work was brought to

Europe by the Jewish mathematician and astronomerAbraham bar Hiyya

(also known as Savasorda) who lived in Barcelona around 1100.

By 1545 ,Girolamo Cardanohad blended the algebra of Al-Khwarizmi

with the Euclidean geometry. His work allowed for the existence of

complex or imaginary roots, as well as negative and irrational roots.

At the end of the 16 th Century the mathematical notation and symbolism

was introduced byFranc ̧ois Viete` in France.

In 1637 , whenRene Descartes ́ publishedLa G ́eometrie ́ , the quadratic

formula adopted the form we see today.

If ax^2 +bx+c=0, a 6 =0, then x=

¡b§

p

b^2 ¡ 4 ac

2 a

.

Proof: If ax^2 +bx+c=0, a 6 =0

then x^2 +

b

a

x+

c

a

=0 fdividing each term bya,asa 6 =0g

) x^2 +

b

a

x =¡

c

a

) x^2 +

b

a

x+

³

b

2 a

́ 2

=¡

c

a

+

³

b

2 a

́ 2

fcompleting the square on LHSg

)

³

x+

b

2 a

́ 2

=

b^2 ¡ 4 ac

4 a^2

ffactorisingg

) x+

b

2 a

=§

r

b^2 ¡ 4 ac

4 a^2

) x=

¡b§

p

b^2 ¡ 4 ac

2 a

From the name

Al-Khwarizmi we get

the word ‘algorithm’.

Muhammad Al-Khwarizmi

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_03\070CamAdd_03.cdr Friday, 31 January 2014 5:02:46 PM BRIAN