EXTREAM ALGEBRA OF MATH 2014

BISMILLAH (Salam Math Lovers)

EXTREAM OF MATH 2014

If bc – k = a^2 and ca – k = b^2, prove that c^2 = ab – k 

PLEASE SOLVE IT...

Solution :

 

We must divide in two cases.

Cases 1.

bc – k = a^2

ca – k = b^2 + (elimination)

bc + ca – 2k = a^2 + b^2

c(a + b) = a^2 + b^2 + 2k

c(a + b) = (a + b)^2 – 2ab + 2k

c= [(a + b)^2 – 2ab + 2k]/( (a + b)…get equation  1

Cases 1.

bc – k = a^2

ca – k = b^2  - (elimination)

bc – ca = a^2 – b^2

bc – ca = (a + b) (a – b)

c (b – a) = (a + b) (a – b)

c = [(a + b) (a – b)] / (b – a)

- c = a + b…get equation 2.

And then Subtitute equation 1 dan 2 to get equation 3

c= [(a + b)^2 – 2ab + 2k]/( (a + b)

c = [(-c)^2 – 2(ab – k) ]/ -c

c^2 – 2(ab – k) = - c^2

c^2 + c^2 = 2 (ab – k )

2c^2 = 2 (ab – k)

So finally, we can prove that

c^2 = ab – k

By AWK – Bogor Indonesia

\Cheers and bravo guys/ ^_^