What Does it Equal? Part 1

Algebra Level 2

If x 2 + x − 2 = 3 x^{2} + x^{-2} = 3 , then what does x + x − 1 x + x^{-1} equal for x > 0 x>0 ?

Click here for part 2

3 \sqrt 3 7 \sqrt 7 5 \sqrt 5 7 7 2 \sqrt 2 5 5 2 2 3 3

Jesse Li by Jesse Li , 16, USA

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2 solutions

Jesse Li
Oct 1, 2018

(x+ 1 x \frac{1}{x} )²= x²+2+ 1 x ² \frac{1}{x²} = 3+2= 5

(x+ 1 x \frac{1}{x} )²=5

(x+ 1 x \frac{1}{x} )= ± 5 \pm\sqrt{5}

Since x > 0 x>0 , 5 \sqrt 5 is the only solution

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Chew-Seong Cheong
Dec 12, 2018

Similar solution with @Jesse Li 's

x 2 + x − 2 = 3 x 2 + 1 x 2 = 3 Add 2 on both sides. x 2 + 2 + 1 x 2 = 5 ( x + 1 x ) 2 = 5 Take square root on both sides. ⟹ x + x − 1 = ± 5 \begin{aligned} x^2 + x^{-2} & = 3 \\ x^2 + \frac 1{x^2} & = 3 & \small \color{#3D99F6} \text{Add 2 on both sides.} \\ x^2 + 2 + \frac 1{x^2} & = 5 \\ \left(x+\frac 1x\right)^2 & = 5 & \small \color{#3D99F6} \text{Take square root on both sides.} \\ \implies x+x^{-1} & = \boxed{\pm \sqrt 5} \end{aligned}

2 Helpful 0 Interesting 0 Brilliant 1 Confused

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