Solve (5(x^(2)-9))/((x-2)(x^5)) / (x-3)/(x^(2) 3x-10)

Question

Simplify the expression

\frac{15 x ^{4} - 50 x + 45 x ^{3} - 150}{x ^{6} - 2 x ^{5}}

Show Solution

x = 0, x = 2, x = \frac{3 90}{3}, x = 3

Evaluate

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) ( x ^{5} )} \div \frac{x - 3}{x ^{2} \times 3 x - 10}

To find the excluded values,set the denominators equal to 0

(x - 2) (x^{5}) = 0 x^{2} \times 3 x - 10 = 0 \frac{x - 3}{x ^{2} \times 3 x - 10} = 0

Solve the equations

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

Solve the equations

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x = 0 x = 2 x = \frac{3 90}{3} \frac{x - 3}{x ^{2} \times 3 x - 10} = 0

Solve the equations

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x = 0 x = 2 x = \frac{3 90}{3} x = 3

Solution

x = 0, x = 2, x = \frac{3 90}{3}, x = 3

Show Solution

x = - 3

Evaluate

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) ( x ^{5} )} \div \frac{x - 3}{x ^{2} \times 3 x - 10}

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

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) ( x ^{5} )} \div \frac{x - 3}{x ^{2} \times 3 x - 10} = 0

Find the domain

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Evaluate

⎩ ⎨ ⎧ (x - 2) x^{5} \neq = 0 x^{2} \times 3 x - 10 \neq = 0 \frac{x - 3}{x ^{2} \times 3 x - 10} \neq = 0 (x - 2) (x^{5}) \neq = 0

Calculate

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⎩ ⎨ ⎧ x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty) x^{2} \times 3 x - 10 \neq = 0 \frac{x - 3}{x ^{2} \times 3 x - 10} \neq = 0 (x - 2) (x^{5}) \neq = 0

Calculate

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⎩ ⎨ ⎧ x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty) x \neq = \frac{3 90}{3} \frac{x - 3}{x ^{2} \times 3 x - 10} \neq = 0 (x - 2) (x^{5}) \neq = 0

Calculate

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⎩ ⎨ ⎧ x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty) x \neq = \frac{3 90}{3} x \neq = 3 (x - 2) (x^{5}) \neq = 0

Calculate

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⎩ ⎨ ⎧ x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty) x \neq = \frac{3 90}{3} x \neq = 3 x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty)

Simplify

⎩ ⎨ ⎧ x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty) x \neq = \frac{3 90}{3} x \neq = 3

Find the intersection

x \in (- \infty, 0) \cup (0, \frac{3 90}{3}) \cup (\frac{3 90}{3}, 2) \cup (2, 3) \cup (3, + \infty)

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) ( x ^{5} )} \div \frac{x - 3}{x ^{2} \times 3 x - 10} = 0, x \in (- \infty, 0) \cup (0, \frac{3 90}{3}) \cup (\frac{3 90}{3}, 2) \cup (2, 3) \cup (3, + \infty)

Calculate

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) ( x ^{5} )} \div \frac{x - 3}{x ^{2} \times 3 x - 10} = 0

Calculate

\frac{5 ( x ^{2} - 9 )}{( x - 2 ) x ^{5}} \div \frac{x - 3}{x ^{2} \times 3 x - 10} = 0

Multiply

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\frac{5 ( x ^{2} - 9 )}{( x - 2 ) x ^{5}} \div \frac{x - 3}{3 x ^{3} - 10} = 0

Multiply the terms

\frac{5 ( x ^{2} - 9 )}{x ^{5} ( x - 2 )} \div \frac{x - 3}{3 x ^{3} - 10} = 0

Divide the terms

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\frac{5 ( x + 3 ) ( 3 x ^{3} - 10 )}{x ^{5} ( x - 2 )} = 0

Cross multiply

5 (x + 3) (3 x^{3} - 10) = x^{5} (x - 2) \times 0

Simplify the equation

5 (x + 3) (3 x^{3} - 10) = 0

Elimination the left coefficient

(x + 3) (3 x^{3} - 10) = 0

Separate the equation into 2 possible cases

x + 3 = 0 3 x^{3} - 10 = 0

Solve the equation

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x = - 3 3 x^{3} - 10 = 0

Solve the equation

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x = - 3 x = \frac{3 90}{3}

Check if the solution is in the defined range

x = - 3 x = \frac{3 90}{3}, x \in (- \infty, 0) \cup (0, \frac{3 90}{3}) \cup (\frac{3 90}{3}, 2) \cup (2, 3) \cup (3, + \infty)

Solution

x = - 3

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