Solve 3x^2-3-7x^2 12x^2-5x^2

Evaluate

3 x^{2} - 3 - 7 x^{2} \times 12 x^{2} - 5 x^{2}

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

3 x^{2} - 3 - 7 x^{2} \times 12 x^{2} - 5 x^{2} = 0

Multiply

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3 x^{2} - 3 - 84 x^{4} - 5 x^{2} = 0

Subtract the terms

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- 2 x^{2} - 3 - 84 x^{4} = 0

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

- 2 t - 3 - 84 t^{2} = 0

Rewrite in standard form

- 84 t^{2} - 2 t - 3 = 0

Multiply both sides

84 t^{2} + 2 t + 3 = 0

Substitute a= 84,b= 2 and c= 3 into the quadratic formula t = \frac{- b \pm b ^{2} - 4 a c}{2 a}

t = \frac{- 2 \pm 2 ^{2} - 4 \times 84 \times 3}{2 \times 84}

Simplify the expression

t = \frac{- 2 \pm 2 ^{2} - 4 \times 84 \times 3}{168}

Simplify the expression

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t = \frac{- 2 \pm - 1004}{168}

Simplify the radical expression

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t = \frac{- 2 \pm 2 251 \times i}{168}

Separate the equation into 2 possible cases

t = \frac{- 2 + 2 251 \times i}{168} t = \frac{- 2 - 2 251 \times i}{168}

Simplify the expression

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t = - \frac{1}{84} + \frac{251}{84} i t = \frac{- 2 - 2 251 \times i}{168}

Simplify the expression

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t = - \frac{1}{84} + \frac{251}{84} i t = - \frac{1}{84} - \frac{251}{84} i

Substitute back

x^{2} = - \frac{1}{84} + \frac{251}{84} i x^{2} = - \frac{1}{84} - \frac{251}{84} i

Solve the equation for x

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Substitute back

x^{2} = - \frac{1}{84} + \frac{251}{84} i

Simplify

x = - \frac{1}{84} + \frac{251}{84} i

Rewrite the complex number in polar form

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x = \frac{7}{14} (cos (arctan (- 251) + π) + i sin (arctan (- 251) + 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 = \frac{7}{14} \times cos \frac{arctan ( - 251 ) + π + 2 kπ}{2} + i sin \frac{arctan ( - 251 ) + π + 2 kπ}{2}

Simplify

x = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π + 2 kπ}{2} + i sin \frac{arctan ( - 251 ) + π + 2 kπ}{2}

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

x_{1} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π + 2 \times 0 \times π}{2} + i sin \frac{arctan ( - 251 ) + π + 2 \times 0 \times π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( - 251 ) + π + 2 \times 1 \times π}{2}

Calculate

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x_{1} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π}{2} + i sin \frac{arctan ( - 251 ) + π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( - 251 ) + π + 2 \times 1 \times π}{2}

Calculate

x_{1} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π}{2} + i sin \frac{arctan ( - 251 ) + π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( - 251 ) + π + 2 π}{2} + i sin \frac{arctan ( - 251 ) + π + 2 π}{2}

Calculate

x_{1} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} i x_{2} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} i

x_{1} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} i x_{2} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} i x^{2} = - \frac{1}{84} - \frac{251}{84} i

Solve the equation for x

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Substitute back

x^{2} = - \frac{1}{84} - \frac{251}{84} i

Simplify

x = - \frac{1}{84} - \frac{251}{84} i

Rewrite the complex number in polar form

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x = \frac{7}{14} (cos (arctan (251) + π) + i sin (arctan (251) + 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 = \frac{7}{14} \times cos \frac{arctan ( 251 ) + π + 2 kπ}{2} + i sin \frac{arctan ( 251 ) + π + 2 kπ}{2}

Simplify

x = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π + 2 kπ}{2} + i sin \frac{arctan ( 251 ) + π + 2 kπ}{2}

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

x_{1} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π + 2 \times 0 \times π}{2} + i sin \frac{arctan ( 251 ) + π + 2 \times 0 \times π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( 251 ) + π + 2 \times 1 \times π}{2}

Calculate

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x_{1} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π}{2} + i sin \frac{arctan ( 251 ) + π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π + 2 \times 1 \times π}{2} + i sin \frac{arctan ( 251 ) + π + 2 \times 1 \times π}{2}

Calculate

x_{1} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π}{2} + i sin \frac{arctan ( 251 ) + π}{2} x_{2} = \frac{4 1372}{14} cos \frac{arctan ( 251 ) + π + 2 π}{2} + i sin \frac{arctan ( 251 ) + π + 2 π}{2}

Calculate

x_{1} = - \frac{4 1372 \times cos ( \frac{π - a r c t a n ( 251 )}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( 251 ) + π}{2} )}{14} i x_{2} = \frac{4 1372 \times cos ( \frac{a r c t a n ( 251 ) + π + 2 π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( 251 ) + π + 2 π}{2} )}{14} i

x_{1} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π}{2} )}{14} i x_{2} = \frac{4 1372 \times cos ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( - 251 ) + π + 2 π}{2} )}{14} i x_{1} = - \frac{4 1372 \times cos ( \frac{π - a r c t a n ( 251 )}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( 251 ) + π}{2} )}{14} i x_{2} = \frac{4 1372 \times cos ( \frac{a r c t a n ( 251 ) + π + 2 π}{2} )}{14} + \frac{4 1372 \times sin ( \frac{a r c t a n ( 251 ) + π + 2 π}{2} )}{14} i

Solution

x \in / R

Simplify the expression