Question

Find the value of x

Answer 1

The equation is true only if both the powers are 1,

5+2√6+5-2√6=10

So if the powers are 1,

x²-3=1

x²=4

x=+2 or -2

Answer 2

Let : a = 5 + 2√6 ... so that ... 5 - 2√6 = 1/a. 

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 .................. Ans. 
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