Solve x^6-9x^4-81x^2 729

Question

Simplify the expression

x^{6} - 9 x^{4} - 59049 x^{2}

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x^{2} (x^{4} - 9 x^{2} - 59049)

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x_{1} = - \frac{3 2 + 2 2917}{2}, x_{2} = 0, x_{3} = \frac{3 2 + 2 2917}{2}

Alternative Form

x_{1} \approx - 15.733457, x_{2} = 0, x_{3} \approx 15.733457

Evaluate

x^{6} - 9 x^{4} - 81 x^{2} \times 729

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

x^{6} - 9 x^{4} - 81 x^{2} \times 729 = 0

Multiply the terms

x^{6} - 9 x^{4} - 59049 x^{2} = 0

Factor the expression

x^{2} (x^{4} - 9 x^{2} - 59049) = 0

Separate the equation into 2 possible cases

x^{2} = 0 x^{4} - 9 x^{2} - 59049 = 0

The only way a power can be 0 is when the base equals 0

x = 0 x^{4} - 9 x^{2} - 59049 = 0

Solve the equation

More Steps

Evaluate

x^{4} - 9 x^{2} - 59049 = 0

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

t^{2} - 9 t - 59049 = 0

Substitute a= 1,b= - 9 and c= - 59049 into the quadratic formula t = \frac{- b \pm b ^{2} - 4 a c}{2 a}

t = \frac{9 \pm ( - 9 ) ^{2} - 4 ( - 59049 )}{2}

Simplify the expression

More Steps

t = \frac{9 \pm 236277}{2}

Simplify the radical expression

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t = \frac{9 \pm 9 2917}{2}

Separate the equation into 2 possible cases

t = \frac{9 + 9 2917}{2} t = \frac{9 - 9 2917}{2}

Substitute back

x^{2} = \frac{9 + 9 2917}{2} x^{2} = \frac{9 - 9 2917}{2}

Solve the equation for x

More Steps

x = \frac{3 2 + 2 2917}{2} x = - \frac{3 2 + 2 2917}{2} x^{2} = \frac{9 - 9 2917}{2}

Solve the equation for x

More Steps

x = \frac{3 2 + 2 2917}{2} x = - \frac{3 2 + 2 2917}{2} x = \frac{3 2 2917 - 2}{2} i x = - \frac{3 2 2917 - 2}{2} i

x = 0 x = \frac{3 2 + 2 2917}{2} x = - \frac{3 2 + 2 2917}{2}

Solution

x_{1} = - \frac{3 2 + 2 2917}{2}, x_{2} = 0, x_{3} = \frac{3 2 + 2 2917}{2}

Alternative Form

x_{1} \approx - 15.733457, x_{2} = 0, x_{3} \approx 15.733457

Show Solution