Solve x^2 - 12x^6 - x = 6 - x - 35

Evaluate

x^{2} - 12 x^{6} - x = 6 - x - 35

Cancel equal terms on both sides of the expression

x^{2} - 12 x^{6} = 6 - 35

Subtract the numbers

x^{2} - 12 x^{6} = - 29

Move the expression to the left side

x^{2} - 12 x^{6} - (- 29) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

x^{2} - 12 x^{6} + 29 = 0

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

t - 12 t^{3} + 29 = 0

Calculate

t \approx 1.362656 t \approx - 0.681328 + 1.144242 i t \approx - 0.681328 - 1.144242 i

Substitute back

x^{2} \approx 1.362656 x^{2} \approx - 0.681328 + 1.144242 i x^{2} \approx - 0.681328 - 1.144242 i

Solve the equation for x

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x \approx 1.167328 x \approx - 1.167328 x^{2} \approx - 0.681328 + 1.144242 i x^{2} \approx - 0.681328 - 1.144242 i

Solve the equation for x

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Substitute back

x^{2} \approx - 0.681328 + 1.144242 i

Simplify

x = - 0.681328 + 1.144242 i

Rewrite the complex number in polar form

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x = 1.331727 (cos (arctan (- 1.679429) + π) + i sin (arctan (- 1.679429) + 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 = 1.331727 \times (cos (\frac{arctan ( - 1.679429 ) + π + 2 kπ}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 kπ}{2}))

Simplify

x = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π + 2 kπ}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 kπ}{2}))

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

x_{1} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π + 2 \times 0 \times π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 \times 0 \times π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π + 2 \times 1 \times π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 \times 1 \times π}{2}))

Calculate

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x_{1} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π + 2 \times 1 \times π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 \times 1 \times π}{2}))

Calculate

x_{1} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( - 1.679429 ) + π + 2 π}{2}) + i sin (\frac{arctan ( - 1.679429 ) + π + 2 π}{2}))

Calculate

x_{1} \approx 0.570263 + 1.003258 i x_{2} \approx - 0.570263 - 1.003258 i

x \approx 1.167328 x \approx - 1.167328 x_{1} \approx 0.570263 + 1.003258 i x_{2} \approx - 0.570263 - 1.003258 i x^{2} \approx - 0.681328 - 1.144242 i

Solve the equation for x

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Substitute back

x^{2} \approx - 0.681328 - 1.144242 i

Simplify

x = - 0.681328 - 1.144242 i

Rewrite the complex number in polar form

More Steps

x = 1.331727 (cos (arctan (1.679429) + π) + i sin (arctan (1.679429) + 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 = 1.331727 \times (cos (\frac{arctan ( 1.679429 ) + π + 2 kπ}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 kπ}{2}))

Simplify

x = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π + 2 kπ}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 kπ}{2}))

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

x_{1} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π + 2 \times 0 \times π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 \times 0 \times π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π + 2 \times 1 \times π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 \times 1 \times π}{2}))

Calculate

More Steps

x_{1} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π + 2 \times 1 \times π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 \times 1 \times π}{2}))

Calculate

x_{1} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π}{2})) x_{2} = 1.154005 (cos (\frac{arctan ( 1.679429 ) + π + 2 π}{2}) + i sin (\frac{arctan ( 1.679429 ) + π + 2 π}{2}))

Calculate

x_{1} \approx - 0.570263 + 1.003258 i x_{2} \approx 0.570263 - 1.003258 i

x \approx 1.167328 x \approx - 1.167328 x_{1} \approx 0.570263 + 1.003258 i x_{2} \approx - 0.570263 - 1.003258 i x_{1} \approx - 0.570263 + 1.003258 i x_{2} \approx 0.570263 - 1.003258 i

Calculate

x \approx 1.167328 x \approx - 1.167328

Solution

x_{1} \approx - 1.167328, x_{2} \approx 1.167328

Solve the equation

Graph