Question :

frac17x-53x^2-2x-15

Find the excluded values

x = 5, x = - 3

Evaluate

\frac{17 x - 53}{x ^{2} - 2 x - 15}

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

x^{2} - 2 x - 15 = 0

Factor the expression

More Steps

Evaluate

x^{2} - 2 x - 15

Rewrite the expression

x^{2} + (3 - 5) x - 15

Calculate

x^{2} + 3 x - 5 x - 15

Rewrite the expression

x \times x + x \times 3 - 5 x - 5 \times 3

Factor out x from the expression

x (x + 3) - 5 x - 5 \times 3

Factor out - 5 from the expression

x (x + 3) - 5 (x + 3)

Factor out x + 3 from the expression

(x - 5) (x + 3)

(x - 5) (x + 3) = 0

When the product of factors equals 0,at least one factor is 0

x - 5 = 0 x + 3 = 0

Solve the equation for x

More Steps

Evaluate

x - 5 = 0

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

x = 0 + 5

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

x = 5

x = 5 x + 3 = 0

Solve the equation for x

More Steps

Evaluate

x + 3 = 0

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

x = 0 - 3

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

x = - 3

x = 5 x = - 3

Solution

x = 5, x = - 3

Show Solution

Rewrite the fraction

\frac{4}{x - 5} + \frac{13}{x + 3}

Evaluate

\frac{17 x - 53}{x ^{2} - 2 x - 15}

Factor the expression

More Steps

Evaluate

x^{2} - 2 x - 15

Rewrite the expression

x^{2} + (3 - 5) x - 15

Calculate

x^{2} + 3 x - 5 x - 15

Rewrite the expression

x \times x + x \times 3 - 5 x - 5 \times 3

Factor out x from the expression

x (x + 3) - 5 x - 5 \times 3

Factor out - 5 from the expression

x (x + 3) - 5 (x + 3)

Factor out x + 3 from the expression

(x - 5) (x + 3)

\frac{17 x - 53}{( x - 5 ) ( x + 3 )}

For each factor in the denominator,write a new fraction

\frac{?}{x - 5} + \frac{?}{x + 3}

Write the terms in the numerator

\frac{A}{x - 5} + \frac{B}{x + 3}

Set the sum of fractions equal to the original fraction

\frac{17 x - 53}{( x - 5 ) ( x + 3 )} = \frac{A}{x - 5} + \frac{B}{x + 3}

Multiply both sides

\frac{17 x - 53}{( x - 5 ) ( x + 3 )} \times (x - 5) (x + 3) = \frac{A}{x - 5} \times (x - 5) (x + 3) + \frac{B}{x + 3} \times (x - 5) (x + 3)

Simplify the expression

17 x - 53 = (x + 3) A + (x - 5) B

Simplify the expression

More Steps

Evaluate

(x + 3) A + (x - 5) B

Multiply the terms

A (x + 3) + (x - 5) B

Multiply the terms

A (x + 3) + B (x - 5)

Expand the expression

A x + 3 A + B (x - 5)

Expand the expression

A x + 3 A + B x - 5 B

17 x - 53 = A x + 3 A + B x - 5 B

Group the terms

17 x - 53 = (A + B) x + 3 A - 5 B

Equate the coefficients

{17 = A + B - 53 = 3 A - 5 B

Swap the sides

{A + B = 17 3 A - 5 B = - 53

Solve the equation for A

{A = 17 - B 3 A - 5 B = - 53

Substitute the given value of A into the equation 3 A - 5 B = - 53

3 (17 - B) - 5 B = - 53

Simplify

More Steps

Evaluate

3 (17 - B) - 5 B

Expand the expression

51 - 3 B - 5 B

Subtract the terms

51 - 8 B

51 - 8 B = - 53

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

- 8 B = - 53 - 51

Subtract the numbers

- 8 B = - 104

Change the signs on both sides of the equation

8 B = 104

Divide both sides

\frac{8 B}{8} = \frac{104}{8}

Divide the numbers

B = \frac{104}{8}

Divide the numbers

More Steps

Evaluate

\frac{104}{8}

Reduce the numbers

\frac{13}{1}

Calculate

13

B = 13

Substitute the given value of B into the equation A = 17 - B

A = 17 - 13

Calculate

A = 4

Calculate

{A = 4 B = 13

Solution

\frac{4}{x - 5} + \frac{13}{x + 3}

Show Solution

Find the roots

x = \frac{53}{17}

Alternative Form

x \approx 3.117647

Evaluate

\frac{17 x - 53}{x ^{2} - 2 x - 15}

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

\frac{17 x - 53}{x ^{2} - 2 x - 15} = 0

Find the domain

More Steps

Evaluate

x^{2} - 2 x - 15 \neq = 0

Move the constant to the right side

x^{2} - 2 x \neq = 0 - (- 15)

Add the terms

x^{2} - 2 x \neq = 15

Add the same value to both sides

x^{2} - 2 x + 1 \neq = 15 + 1

Evaluate

x^{2} - 2 x + 1 \neq = 16

Evaluate

(x - 1)^{2} \neq = 16

Take the root of both sides of the equation and remember to use both positive and negative roots

x - 1 \neq = \pm 16

Simplify the expression

More Steps

Evaluate

16

Write the number in exponential form with the base of 4

4^{2}

Reduce the index of the radical and exponent with 2

4

x - 1 \neq = \pm 4

Separate the inequality into 2 possible cases

{x - 1 \neq = 4 x - 1 \neq = - 4

Calculate

More Steps

Evaluate

x - 1 \neq = 4

Move the constant to the right side

x \neq = 4 + 1

Add the numbers

x \neq = 5

{x \neq = 5 x - 1 \neq = - 4

Calculate

More Steps

Evaluate

x - 1 \neq = - 4

Move the constant to the right side

x \neq = - 4 + 1

Add the numbers

x \neq = - 3

{x \neq = 5 x \neq = - 3

Find the intersection

x \in (- \infty, - 3) \cup (- 3, 5) \cup (5, + \infty)

\frac{17 x - 53}{x ^{2} - 2 x - 15} = 0, x \in (- \infty, - 3) \cup (- 3, 5) \cup (5, + \infty)

Calculate

\frac{17 x - 53}{x ^{2} - 2 x - 15} = 0

Cross multiply

17 x - 53 = (x^{2} - 2 x - 15) \times 0

Simplify the equation

17 x - 53 = 0

Move the constant to the right side

17 x = 0 + 53

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

17 x = 53

Divide both sides

\frac{17 x}{17} = \frac{53}{17}

Divide the numbers

x = \frac{53}{17}

Check if the solution is in the defined range

x = \frac{53}{17}, x \in (- \infty, - 3) \cup (- 3, 5) \cup (5, + \infty)

Solution

x = \frac{53}{17}

Alternative Form

x \approx 3.117647

Show Solution

frac17x-53x^2-2x-15

Find the excluded values

Rewrite the fraction

Find the roots

Frequently Asked Questions

Find the excluded values of \(\frac{17x-53}{x^2-2x-15}\)

Rewrite using partial-fraction decomposition of \(\frac{17x-53}{x^2-2x-15}\)

Find the roots of the algebra expression of \(\frac{17x-53}{x^2-2x-15}\)