Solve (a- 1/a -3 )(a^2 1/a^2 10 - 3/a^3a)

Question

Simplify the expression

\frac{10 a ^{4} - 13 a ^{2} + 3 - 30 a ^{3} + 9 a}{a ^{3}}

Evaluate

(a - \frac{1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a)

Subtract the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a)

Multiply the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (10 - \frac{3}{a ^{3}} \times a)

Multiply the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (10 - \frac{3}{a ^{2}})

Subtract the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times \frac{10 a ^{2} - 3}{a ^{2}}

Multiply the terms

\frac{( a ^{2} - 1 - 3 a ) ( 10 a ^{2} - 3 )}{a \times a ^{2}}

Multiply the terms

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\frac{( a ^{2} - 1 - 3 a ) ( 10 a ^{2} - 3 )}{a ^{3}}

Solution

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\frac{10 a ^{4} - 13 a ^{2} + 3 - 30 a ^{3} + 9 a}{a ^{3}}

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Find the excluded values

a = 0

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Find the roots

a_{1} = - \frac{30}{10}, a_{2} = \frac{3 - 13}{2}, a_{3} = \frac{30}{10}, a_{4} = \frac{3 + 13}{2}

Alternative Form

a_{1} \approx - 0.547723, a_{2} \approx - 0.302776, a_{3} \approx 0.547723, a_{4} \approx 3.302776

Evaluate

(a - \frac{1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a)

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

(a - \frac{1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a) = 0

Find the domain

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(a - \frac{1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a) = 0, a \neq = 0

Calculate

(a - \frac{1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a) = 0

Subtract the terms

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(\frac{a ^{2} - 1}{a} - 3) (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a) = 0

Subtract the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (a^{2} \times \frac{1}{a ^{2}} \times 10 - \frac{3}{a ^{3}} \times a) = 0

Multiply the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (10 - \frac{3}{a ^{3}} \times a) = 0

Multiply the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times (10 - \frac{3}{a ^{2}}) = 0

Subtract the terms

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\frac{a ^{2} - 1 - 3 a}{a} \times \frac{10 a ^{2} - 3}{a ^{2}} = 0

Multiply the terms

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\frac{( a ^{2} - 1 - 3 a ) ( 10 a ^{2} - 3 )}{a ^{3}} = 0

Cross multiply

(a^{2} - 1 - 3 a) (10 a^{2} - 3) = a^{3} \times 0

Simplify the equation

(a^{2} - 1 - 3 a) (10 a^{2} - 3) = 0

Separate the equation into 2 possible cases

a^{2} - 1 - 3 a = 0 10 a^{2} - 3 = 0

Solve the equation

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a = \frac{3 + 13}{2} a = \frac{3 - 13}{2} 10 a^{2} - 3 = 0

Solve the equation

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a = \frac{3 + 13}{2} a = \frac{3 - 13}{2} a = \frac{30}{10} a = - \frac{30}{10}

Check if the solution is in the defined range

a = \frac{3 + 13}{2} a = \frac{3 - 13}{2} a = \frac{30}{10} a = - \frac{30}{10}, a \neq = 0

Find the intersection of the solution and the defined range

a = \frac{3 + 13}{2} a = \frac{3 - 13}{2} a = \frac{30}{10} a = - \frac{30}{10}

Solution

a_{1} = - \frac{30}{10}, a_{2} = \frac{3 - 13}{2}, a_{3} = \frac{30}{10}, a_{4} = \frac{3 + 13}{2}

Alternative Form

a_{1} \approx - 0.547723, a_{2} \approx - 0.302776, a_{3} \approx 0.547723, a_{4} \approx 3.302776

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