How many zeroes

Algebra Level pending

The equation x 4 − x 2 + 2 x − 1 = 0 x^{4}-x^{2}+2x-1=0 has n n real zeroes. What is n n ?

4 0 2

Priyansh Singh by PRIYANSH SINGH , 19, India

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2 solutions

Expanding from @PRIYANSH SINGH 's solution.

x 4 − x 2 + 2 x − 1 = 0 Rearranging x 4 = x 2 − 2 x + 1 x 4 = ( x − 1 ) 2 ⟹ x 2 = ± ( x − 1 ) \begin{aligned} x^4 - x^2 + 2x - 1 & = 0 & \small \color{#3D99F6} \text{Rearranging} \\ x^4 & = x^2 - 2x + 1 \\ x^4 & = (x-1)^2 \\ \implies x^2 & = \pm (x-1) \end{aligned}

⟹ { x 2 − x + 1 = 0 No real roots x 2 + x − 1 = 0 ⟹ x = ± 5 − 1 2 2 real roots \implies \begin{cases} x^2 - x + 1 = 0 & & \color{#D61F06} \text{No real roots} \\ x^2 + x - 1 = 0 & \implies x = \dfrac {\pm\sqrt 5-1}2 & \color{#3D99F6} \boxed 2 \text{ real roots} \end{cases}

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Priyansh Singh
Nov 9, 2018

The above equation can be written as x 4 x^{4} - ( x − 1 ) 2 (x-1)^{2} =0

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@PRIYANSH SINGH , just use a pair of \ ( \ ) for the entire equation. You can see the LaTex code by placing the mouse cursor on top of the formulas.

Chew-Seong Cheong - 2 years, 7 months ago

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