What are the roots of $x^{3} - 8 x - 3 = 0$ $x^3-8x-3=0$ ?

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What are the roots of $x^{3} - 8 x - 3 = 0$ $x^3-8x-3=0$ ?

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Answer 1 · 2017-03-24T16:53:11

Three roots of $x^{3} - 8 x - 3 = 0$ $x^3-8x-3=0$ are $3$ $3$ , $\frac{- 3 + \sqrt{5}}{2}$ $(-3+sqrt5)/2$ and $\frac{- 3 - \sqrt{5}}{2}$ $(-3-sqrt5)/2$

Explanation:

in an equation of type $x^n+a_1x^(n-1)+a_2x^(n-2)+.........+a_n$ , the roots of the equation are factors of $a_n$ .

Here, we have the equation as $x^3-8x-3=0$ and hence roots are factors of $3$ i.e. $+_1$ and $+-3$ .

It is apparent that $x=3$ is a root as it satisfies the equation

$3^3-8xx3-3=27-24-3=0$ and hence

$x-3$ is a factor of $x^3-8x-3$ and dividing latter by former

$x^2(x-3)+3x(x-3)+1(x-3)=(x-3)(x^2+3x+1)$

As such we have $(x-3)(x^2+3x+1)=0$

And as $x^2+3x+1$ cannot be factorized as rational factors, using quadratic formula roots are

$(-3+-sqrt(3^2-4xx1xx1))/2=(-3+-sqrt5)/2$

and hence three roots of $x^3-8x-3=0$ are $3$ , $(-3+sqrt5)/2$ and $(-3-sqrt5)/2$

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What are the roots of $x^{3} - 8 x - 3 = 0$ $x^3-8x-3=0$ ?

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Explanation:

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What are the roots of x^3-8x-3=0x3−8x−3=0x^3-8x-3=0?

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Explanation:

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What are the roots of $x^{3} - 8 x - 3 = 0$ $x^3-8x-3=0$ ?