Question

If a(2+√3)=b(2-√3). Then find the value of ab .

Answer 1

this equation can be written as

a(2+√3) - b(2-√3) = o

now we squaring on both side

[a(2+√3) - b(2-√3)]^2 = 0

this is in the form of (ax-by)^2

= a^2x^2 +b^2y^2 - 2*ax * by

where ax = a * (2+√3)

and by = b*(2-√3)

[a(2+√3) - b(2-√3)]^2

= a^2 * [(2+√3)]^2 + b^2 * [(2-√3)]^2 - 2*a*b* (2+√3)*(2-√3) = 0

we take ab containing term in the lhs


2 a*b* (2+√3)*(2-√3) = a^2 * [(2+√3)]^2 + b^2 * [(2-√3)]^2

we know that ( a+b ) * (a-b) = a^2-b^2

so
2*a*b *[ 2^2 - √3^2 ] = a^2 * [(2+√3)]^2 + b^2 * [(2-√3)]^2

= 2ab [ 4-3 ] = a^2 * [(2+√3)]^2 + b^2 * [(2-√3)]^2

ab = {a^2 * [(2+√3)]^2 + b^2 * [(2-√3)]^2} / 2