How many of 'em fits in well?

Algebra Level 3

If x 3 + x = 0 x^3 + x = 0 for some real number x x , then how many values of x x satisfy this equation?

0 3 1 2

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2 solutions

Ashish Menon
Sep 25, 2016

x 3 + x = 0 x ( x 2 + 1 ) = 0 x = 0 ( or ) x 2 + 1 = 0 x = 0 ( or ) x 2 = 1 But for real number x , x 2 0 x 2 1 x = 0 \begin{aligned} x^3 + x & = 0\\ \\ x(x^2 + 1) & = 0\\ \\ x = 0 & (\text{or}) x^2 + 1 = 0\\ \\ x = 0 & (\text{or}) x^2 = -1\\ \\ \text{But for real number x}, x^2 & \geq 0\\ \\ \implies x^2 & \neq -1\\ \\ \implies x & = 0 \end{aligned}

Thus only 1 \color{#3D99F6}{\boxed{1}} value satisfies this equation.

Viki Zeta
Sep 25, 2016

x 3 + x = 0 x ( x 2 + 1 ) = 0 x = 0 , x 2 = 1 x = 0 , x = i are solutions to the expression x^3+x=0\\ x(x^2+1)=0\\ \implies x=0, x^2=-1\\ \therefore x=0, x=i \text{ are solutions to the expression}

Here x = 0 x=0 is real whereas x = i x=i is imaginary so it have 1 real solution

It's a cubic equation and you have only two solutions!

Actually, there will be three roots x = 0 x=0 and x = ± i x= \pm i .You forgot that x = i x=-i is also a root to this equation.

Tapas Mazumdar - 4 years, 8 months ago

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That won't affect the answer of the question. A cubic root has atmost 3 solutions. I agree i -i is also a soln

Viki Zeta - 4 years, 8 months ago

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