Question

solve the cubic equation
x3-4x=53117

Answer 1

Step-by-step explanation:

x3-4x= 53117

3x-4x= 53117

-x= 53117

x= -53117

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Answer 2

To find :

values of x such that:

$f(x) = {x}^{3} - 4x - 53117 = 0$

Solution:

Before finding the roots, let us find the nature roots.

So, We must calculate the

Discriminant [f(x),x]

Δ = –27(–53117)²–4(–4)³ = -76178223347 < 0

as, Δ < 0 ⇔ x₁ ∈ ℝ and, x₂,₃ ∉ ℝ

As, there exists only one real root, so we must use the algebraic cubic formula for the roots.

$x_{1} = \frac{ \omega _{k}. \sqrt[3]{ - 108( - 53117) + 12\sqrt{3(76178223347)} } + \omega _{k} {}^{2} . \sqrt[3]{ - 108( - 53117) - 12 \sqrt{3(76178223347)} } }{6}$

After Simplification , all the roots are

$x_{1} = \frac{ \sqrt[3]{5736636 + 12 \sqrt{228534670041} } + \sqrt[3]{5736636 - 12 \sqrt{228534670041} } }{6}$

$x _{2,3} = \frac{ - \sqrt[3]{5736636 + 12 \sqrt{228534670041} } - \sqrt[3]{5736636 - 12 \sqrt{228534670041} } + \sqrt{3} i( ±{ \sqrt[3]{5736636 + 12 \sqrt{228534670041} } ∓ \sqrt[3]{5736636 - 12 \sqrt{228534670041} } }) }{12}$

where, i = √(–1)

and, ∛(z) represents the cube root.