If (5+2 root6)x2-3+(5-2 root6)x2-3=10, then x=
Dear student,
Let : a = 5 + 2√6 ... so that ... 5 - 2√6 = 1/a. (by rationalization method)
Also, let : n = x² - 3.
Then, the given equation becomes :
... aⁿ + ( 1/ aⁿ ) = 10
∴ (aⁿ)² + 1 = 10(aⁿ)
∴ 1(aⁿ)² - 10(aⁿ) + 1 = 0 ... like ax²+bx+c = 0
∴ by Quadratic Formula,
... aⁿ = [ -b ± √(b²-4ac) ] / (2a) = [ 10 ± √(100- 4) ] / 2
∴ aⁿ = [ 10 ± 4√6 ] / 2 = 5 ± 2√6
∴ ( 5 + 2√6 )ⁿ = ( 5 + 2√6 ) or ( 5 - 2√6) ∴ n = 1 or -1
∴ x² - 3 = 1 or -1
∴ x² = 4 or 2
∴ x = ± 2 or ± √2
Regards
Let : a = 5 + 2√6 ... so that ... 5 - 2√6 = 1/a. (by rationalization method)
Also, let : n = x² - 3.
Then, the given equation becomes :
... aⁿ + ( 1/ aⁿ ) = 10
∴ (aⁿ)² + 1 = 10(aⁿ)
∴ 1(aⁿ)² - 10(aⁿ) + 1 = 0 ... like ax²+bx+c = 0
∴ by Quadratic Formula,
... aⁿ = [ -b ± √(b²-4ac) ] / (2a) = [ 10 ± √(100- 4) ] / 2
∴ aⁿ = [ 10 ± 4√6 ] / 2 = 5 ± 2√6
∴ ( 5 + 2√6 )ⁿ = ( 5 + 2√6 ) or ( 5 - 2√6) ∴ n = 1 or -1
∴ x² - 3 = 1 or -1
∴ x² = 4 or 2
∴ x = ± 2 or ± √2
Regards