Solve x^4-5x^2+40

Evaluate

x^{4} - 5 x^{2} + 40

To find the roots of the expression,set the expression equal to 0

x^{4} - 5 x^{2} + 40 = 0

Solve the equation using substitution t = x^{2}

t^{2} - 5 t + 40 = 0

Substitute a= 1,b= - 5 and c= 40 into the quadratic formula t = \frac{- b \pm b ^{2} - 4 a c}{2 a}

t = \frac{5 \pm ( - 5 ) ^{2} - 4 \times 40}{2}

Simplify the expression

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t = \frac{5 \pm - 135}{2}

Simplify the radical expression

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t = \frac{5 \pm 3 15 \times i}{2}

Separate the equation into 2 possible cases

t = \frac{5 + 3 15 \times i}{2} t = \frac{5 - 3 15 \times i}{2}

Simplify the expression

t = \frac{5}{2} + \frac{3 15}{2} i t = \frac{5 - 3 15 \times i}{2}

Simplify the expression

t = \frac{5}{2} + \frac{3 15}{2} i t = \frac{5}{2} - \frac{3 15}{2} i

Substitute back

x^{2} = \frac{5}{2} + \frac{3 15}{2} i x^{2} = \frac{5}{2} - \frac{3 15}{2} i

Solve the equation for x

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Substitute back

x^{2} = \frac{5}{2} + \frac{3 15}{2} i

Simplify

x = \frac{5}{2} + \frac{3 15}{2} i

Rewrite the complex number in polar form

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x = 210 \times cos arctan \frac{\frac{3 15}{2}}{\frac{5}{2}} + i sin arctan \frac{\frac{3 15}{2}}{\frac{5}{2}}

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

x = 210 \times cos \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 kπ}{2} + i sin \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 kπ}{2}

Simplify

x = 440 \times cos \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 kπ}{2} + i sin \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 kπ}{2}

Since n = 2,substitute k = 0, 1 into the expression

x_{1} = 440 \times cos \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 0 \times π}{2} + i sin \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 0 \times π}{2} x_{2} = 440 \times cos \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 1 \times π}{2} + i sin \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 1 \times π}{2}

Calculate

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x_{1} = 440 \times cos \frac{arctan ( \frac{3 15}{5} )}{2} + i sin \frac{arctan ( \frac{3 15}{5} )}{2} x_{2} = 440 \times cos \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 1 \times π}{2} + i sin \frac{arctan ( \frac{\frac{3 15}{2}}{\frac{5}{2}} ) + 2 \times 1 \times π}{2}

Calculate

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x_{1} = 440 \times cos \frac{arctan ( \frac{3 15}{5} )}{2} + i sin \frac{arctan ( \frac{3 15}{5} )}{2} x_{2} = 440 \times cos \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} + i sin \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2}

Calculate

x_{1} = 440 \times cos \frac{arctan ( \frac{3 15}{5} )}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} )}{2} \times i x_{2} = 440 \times cos \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} \times i

x_{1} = 440 \times cos \frac{arctan ( \frac{3 15}{5} )}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} )}{2} \times i x_{2} = 440 \times cos \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} \times i x^{2} = \frac{5}{2} - \frac{3 15}{2} i

Solve the equation for x

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Substitute back

x^{2} = \frac{5}{2} - \frac{3 15}{2} i

Simplify

x = \frac{5}{2} - \frac{3 15}{2} i

Rewrite the complex number in polar form

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x = 210 \times (cos (arctan (- \frac{3 15}{5}) + 2 π) + i sin (arctan (- \frac{3 15}{5}) + 2 π))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

x = 210 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 kπ}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 kπ}{2}

Simplify

x = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 kπ}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 kπ}{2}

Since n = 2,substitute k = 0, 1 into the expression

x_{1} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 0 \times π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 0 \times π}{2} x_{2} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 1 \times π}{2}

Calculate

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x_{1} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} x_{2} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 \times 1 \times π}{2}

Calculate

x_{1} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} x_{2} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2} + i sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2}

Calculate

x_{1} = - 440 \times cos - \frac{arctan ( - \frac{3 15}{5} )}{2} + 440 \times sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} \times i x_{2} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2} + 440 \times sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2} \times i

x_{1} = 440 \times cos \frac{arctan ( \frac{3 15}{5} )}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} )}{2} \times i x_{2} = 440 \times cos \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} + 440 \times sin \frac{arctan ( \frac{3 15}{5} ) + 2 π}{2} \times i x_{1} = - 440 \times cos - \frac{arctan ( - \frac{3 15}{5} )}{2} + 440 \times sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π}{2} \times i x_{2} = 440 \times cos \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2} + 440 \times sin \frac{arctan ( - \frac{3 15}{5} ) + 2 π + 2 π}{2} \times i

Solution

x \in / R

X^4 5x^2+40