Math Olympiad | A Nice Exponential Problem | VIJAY Maths

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[@uttiyamajumdar8928]

x^(1/4)+x^(1/2)=1 ⇒√t+t=1 [Taking √x=t] ⇒√t=1-t ⇒t=1-2t+t^2 ⇒t^2-3t+1=0 ⇒t=(3±√(9-4))/2=(3±√5)/2 ⇒√x=(3±√5)/2 ⇒x=1/4 (3±√5)^2=1/4 (9+5±6√5)=1/4 (14±6√5) ⇒x=1/2 (7±3√5)

[@ganeshdas3174]

x = 3 + -√5

[@RealQinnMalloryu4]

{x^1x+1x ➖ }/{4x+4x ➖ }+{x^1x+1x ➖ }/{2x+2x ➖ }={2'x^2/8x^2+2x^2/4x^2}=4x^4/12x^4 2^2x^2^2/3^4x^4 1^1x1^1/3^1^1x^1^2 3x^2 (x ➖ 3x+2).

[@walterwen2975]

Math Olympiad: x¹⸍⁴ + x¹⸍² = 1, x ϵ R; x =? 1 > x¹⸍² > x¹⸍⁴ > 0; 1 > x > 0, x ϵ R⁺ x¹⸍² + x¹⸍⁴ - 1 = (x¹⸍⁴)² + x¹⸍⁴ - 1 = 0, x¹⸍⁴ = (- 1 + √5)/2 > 0 x¹⸍² = [(- 1 + √5)/2]² = (- 1 + √5)²/4 = (1 + 5 - 2√5)/4 = (3 - √5)/2 = (√9 - √5)/2 x = [(√9 - √5)/2]² = (14 - 2√45)/4 = (√49 - √45)/2 = (7 - 3√5)/2 Answer check: x = (7 - 3√5)/2; x¹⸍⁴ = (- 1 + √5)/2, x¹⸍² = (3 - √5)/2 x¹⸍⁴ + x¹⸍² = (- 1 + √5)/2 + (3 - √5)/2 = - 1/2 + 3/2 = 1; Confirmed Final answer: x = (7 - 3√5)/2