Rewrite using partial-fraction decomposition of \frac{(3x+5)}{(x(x+2))}

\frac{5}{2x}+\frac{1}{2\left(x+2\right)}

Question :

frac(3x+5)(x(x+2))

Simplify the expression

\frac{3 x + 5}{x ^{2} + 2 x}

Evaluate

\frac{( 3 x + 5 )}{( x ( x + 2 ) )}

Remove the parentheses

\frac{3 x + 5}{x ( x + 2 )}

Solution

More Steps

Evaluate

x (x + 2)

Apply the distributive property

x \times x + x \times 2

Multiply the terms

x^{2} + x \times 2

Use the commutative property to reorder the terms

x^{2} + 2 x

\frac{3 x + 5}{x ^{2} + 2 x}

Show Solution

Find the excluded values

x = 0, x = - 2

Evaluate

\frac{( 3 x + 5 )}{( x ( x + 2 ) )}

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

x (x + 2) = 0

Separate the equation into 2 possible cases

x = 0 x + 2 = 0

Solve the equation

More Steps

Evaluate

x + 2 = 0

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

x = 0 - 2

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

x = - 2

x = 0 x = - 2

Solution

x = 0, x = - 2

Show Solution

Rewrite the fraction

\frac{5}{2 x} + \frac{1}{2 ( x + 2 )}

Evaluate

\frac{( 3 x + 5 )}{( x ( x + 2 ) )}

Remove the parentheses

\frac{3 x + 5}{x ( x + 2 )}

For each factor in the denominator,write a new fraction

\frac{?}{x} + \frac{?}{x + 2}

Write the terms in the numerator

\frac{A}{x} + \frac{B}{x + 2}

Set the sum of fractions equal to the original fraction

\frac{3 x + 5}{x ( x + 2 )} = \frac{A}{x} + \frac{B}{x + 2}

Multiply both sides

\frac{3 x + 5}{x ( x + 2 )} \times x (x + 2) = \frac{A}{x} \times x (x + 2) + \frac{B}{x + 2} \times x (x + 2)

Simplify the expression

3 x + 5 = (x + 2) A + x B

Simplify the expression

More Steps

Evaluate

(x + 2) A + x B

Multiply the terms

A (x + 2) + x B

Expand the expression

A x + 2 A + x B

3 x + 5 = A x + 2 A + x B

Group the terms

3 x + 5 = (A + B) x + 2 A

Equate the coefficients

{3 = A + B 5 = 2 A

Swap the sides

{A + B = 3 2 A = 5

Solve the equation for A

More Steps

Evaluate

2 A = 5

Divide both sides

\frac{2 A}{2} = \frac{5}{2}

Divide the numbers

A = \frac{5}{2}

{A + B = 3 A = \frac{5}{2}

Substitute the given value of A into the equation A + B = 3

\frac{5}{2} + B = 3

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

B = 3 - \frac{5}{2}

Subtract the numbers

More Steps

Evaluate

3 - \frac{5}{2}

Reduce fractions to a common denominator

\frac{3 \times 2}{2} - \frac{5}{2}

Write all numerators above the common denominator

\frac{3 \times 2 - 5}{2}

Multiply the numbers

\frac{6 - 5}{2}

Subtract the numbers

\frac{1}{2}

B = \frac{1}{2}

Calculate

{A = \frac{5}{2} B = \frac{1}{2}

Solution

\frac{5}{2 x} + \frac{1}{2 ( x + 2 )}

Show Solution

Find the roots

x = - \frac{5}{3}

Alternative Form

x = - 1. \dot{6}

Evaluate

\frac{( 3 x + 5 )}{( x ( x + 2 ) )}

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

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

Find the domain

More Steps

Evaluate

x (x + 2) \neq = 0

Apply the zero product property

{x \neq = 0 x + 2 \neq = 0

Solve the inequality

More Steps

Evaluate

x + 2 \neq = 0

Move the constant to the right side

x \neq = 0 - 2

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

x \neq = - 2

{x \neq = 0 x \neq = - 2

Find the intersection

x \in (- \infty, - 2) \cup (- 2, 0) \cup (0, + \infty)

\frac{( 3 x + 5 )}{( x ( x + 2 ) )} = 0, x \in (- \infty, - 2) \cup (- 2, 0) \cup (0, + \infty)

Calculate

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

Remove the parentheses

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

Multiply the terms

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

Cross multiply

3 x + 5 = x (x + 2) \times 0

Simplify the equation

3 x + 5 = 0

Move the constant to the right side

3 x = 0 - 5

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

3 x = - 5

Divide both sides

\frac{3 x}{3} = \frac{- 5}{3}

Divide the numbers

x = \frac{- 5}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x = - \frac{5}{3}

Check if the solution is in the defined range

x = - \frac{5}{3}, x \in (- \infty, - 2) \cup (- 2, 0) \cup (0, + \infty)

Solution

x = - \frac{5}{3}

Alternative Form

x = - 1. \dot{6}

Show Solution

frac(3x+5)(x(x+2))

Simplify the expression

Find the excluded values

Rewrite the fraction

Find the roots

Frequently Asked Questions

Solution of \(\frac{(3x+5)}{(x(x+2))}\)

Find the excluded values of \(\frac{(3x+5)}{(x(x+2))}\)

Rewrite using partial-fraction decomposition of \(\frac{(3x+5)}{(x(x+2))}\)

Find the roots of the algebra expression of \(\frac{(3x+5)}{(x(x+2))}\)