Solve (2x-1)/(2x^(4)-11x^(3) 9x^(2) 7x)

Question

Simplify the expression

\frac{2 x - 1}{2 x ^{4} - 693 x ^{6}}

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Find the excluded values

x = 0, x = \frac{154}{231}, x = - \frac{154}{231}

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Rewrite the fraction

- \frac{1}{2 x ^{4}} + \frac{1}{x ^{3}} - \frac{693}{4 x ^{2}} + \frac{693}{2 x} + \frac{- 480249 + 960498 x}{4 ( 2 - 693 x ^{2} )}

Evaluate

\frac{2 x - 1}{2 x ^{4} - 11 x ^{3} \times 9 x ^{2} \times 7 x}

Evaluate

\frac{2 x - 1}{2 x ^{4} - 693 x ^{6}}

Factor the expression

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\frac{2 x - 1}{x ^{4} ( 2 - 693 x ^{2} )}

For each factor in the denominator,write a new fraction

\frac{?}{x ^{4}} + \frac{?}{x ^{3}} + \frac{?}{x ^{2}} + \frac{?}{x} + \frac{?}{2 - 693 x ^{2}}

Write the terms in the numerator

\frac{A}{x ^{4}} + \frac{B}{x ^{3}} + \frac{C}{x ^{2}} + \frac{D}{x} + \frac{E x + F}{2 - 693 x ^{2}}

Set the sum of fractions equal to the original fraction

\frac{2 x - 1}{x ^{4} ( 2 - 693 x ^{2} )} = \frac{A}{x ^{4}} + \frac{B}{x ^{3}} + \frac{C}{x ^{2}} + \frac{D}{x} + \frac{E x + F}{2 - 693 x ^{2}}

Multiply both sides

\frac{2 x - 1}{x ^{4} ( 2 - 693 x ^{2} )} \times x^{4} (2 - 693 x^{2}) = \frac{A}{x ^{4}} \times x^{4} (2 - 693 x^{2}) + \frac{B}{x ^{3}} \times x^{4} (2 - 693 x^{2}) + \frac{C}{x ^{2}} \times x^{4} (2 - 693 x^{2}) + \frac{D}{x} \times x^{4} (2 - 693 x^{2}) + \frac{E x + F}{2 - 693 x ^{2}} \times x^{4} (2 - 693 x^{2})

Simplify the expression

2 x - 1 = (2 - 693 x^{2}) A + (2 x - 693 x^{3}) B + (2 x^{2} - 693 x^{4}) C + (2 x^{3} - 693 x^{5}) D + x^{4} (E x + F)

Simplify the expression

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2 x - 1 = 2 A - 693 A x^{2} + 2 B x - 693 B x^{3} + 2 C x^{2} - 693 C x^{4} + 2 D x^{3} - 693 D x^{5} + x^{5} E + x^{4} F

Group the terms

2 x - 1 = (- 693 D + E) x^{5} + (- 693 C + F) x^{4} + (- 693 B + 2 D) x^{3} + (- 693 A + 2 C) x^{2} + 2 B x + 2 A

Equate the coefficients

⎩ ⎨ ⎧ 0 = - 693 D + E 0 = - 693 C + F 0 = - 693 B + 2 D 0 = - 693 A + 2 C 2 = 2 B - 1 = 2 A

Swap the sides

⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 B + 2 D = 0 - 693 A + 2 C = 0 2 B = 2 2 A = - 1

Solve the equation for B

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⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 B + 2 D = 0 - 693 A + 2 C = 0 B = 1 2 A = - 1

Substitute the given value of B into the equation ⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 B + 2 D = 0 - 693 A + 2 C = 0 2 A = - 1

⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 \times 1 + 2 D = 0 - 693 A + 2 C = 0 2 A = - 1

Any expression multiplied by 1 remains the same

⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 + 2 D = 0 - 693 A + 2 C = 0 2 A = - 1

Solve the equation for D

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⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 D = \frac{693}{2} - 693 A + 2 C = 0 2 A = - 1

Substitute the given value of D into the equation ⎩ ⎨ ⎧ - 693 D + E = 0 - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

⎩ ⎨ ⎧ - 693 \times \frac{693}{2} + E = 0 - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

Simplify

⎩ ⎨ ⎧ - \frac{480249}{2} + E = 0 - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

Solve the equation for E

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⎩ ⎨ ⎧ E = \frac{480249}{2} - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

Substitute the given value of E into the equation ⎩ ⎨ ⎧ - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

⎩ ⎨ ⎧ - 693 C + F = 0 - 693 A + 2 C = 0 2 A = - 1

Solve the equation for A

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⎩ ⎨ ⎧ - 693 C + F = 0 - 693 A + 2 C = 0 A = - \frac{1}{2}

Substitute the given value of A into the equation {- 693 C + F = 0 - 693 A + 2 C = 0

{- 693 C + F = 0 - 693 (- \frac{1}{2}) + 2 C = 0

Simplify

{- 693 C + F = 0 \frac{693}{2} + 2 C = 0

Solve the equation for C

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{- 693 C + F = 0 C = - \frac{693}{4}

Substitute the given value of C into the equation - 693 C + F = 0

- 693 (- \frac{693}{4}) + F = 0

Multiply the numbers

\frac{480249}{4} + F = 0

Move the constant to the right-hand side and change its sign

F = 0 - \frac{480249}{4}

Removing 0 doesn't change the value,so remove it from the expression

F = - \frac{480249}{4}

Calculate

⎩ ⎨ ⎧ A = - \frac{1}{2} B = 1 C = - \frac{693}{4} D = \frac{693}{2} E = \frac{480249}{2} F = - \frac{480249}{4}

Solution

- \frac{1}{2 x ^{4}} + \frac{1}{x ^{3}} - \frac{693}{4 x ^{2}} + \frac{693}{2 x} + \frac{- 480249 + 960498 x}{4 ( 2 - 693 x ^{2} )}

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Find the roots

x = \frac{1}{2}

Alternative Form

x = 0.5

Evaluate

\frac{2 x - 1}{2 x ^{4} - 11 x ^{3} \times 9 x ^{2} \times 7 x}

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

\frac{2 x - 1}{2 x ^{4} - 11 x ^{3} \times 9 x ^{2} \times 7 x} = 0

Find the domain

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\frac{2 x - 1}{2 x ^{4} - 11 x ^{3} \times 9 x ^{2} \times 7 x} = 0, x \in (- \infty, - \frac{154}{231}) \cup (- \frac{154}{231}, 0) \cup (0, \frac{154}{231}) \cup (\frac{154}{231}, + \infty)

Calculate

\frac{2 x - 1}{2 x ^{4} - 11 x ^{3} \times 9 x ^{2} \times 7 x} = 0

Multiply

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\frac{2 x - 1}{2 x ^{4} - 693 x ^{6}} = 0

Cross multiply

2 x - 1 = (2 x^{4} - 693 x^{6}) \times 0

Simplify the equation

2 x - 1 = 0

Move the constant to the right side

2 x = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

2 x = 1

Divide both sides

\frac{2 x}{2} = \frac{1}{2}

Divide the numbers

x = \frac{1}{2}

Check if the solution is in the defined range

x = \frac{1}{2}, x \in (- \infty, - \frac{154}{231}) \cup (- \frac{154}{231}, 0) \cup (0, \frac{154}{231}) \cup (\frac{154}{231}, + \infty)

Solution

x = \frac{1}{2}

Alternative Form

x = 0.5

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