Precalculus Examples

$\frac{5}{{(x^{2} - 25)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

Factor the fraction.

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Step 1.1.1

Rewrite $25$ as $5^{2}$ .

$\frac{5}{{(x^{2} - 5^{2})}^{2}}$

Step 1.1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\frac{5}{{((x + 5) (x - 5))}^{2}}$

Step 1.1.3

Apply the product rule to $(x + 5) (x - 5)$ .

$\frac{5}{{(x + 5)}^{2} {(x - 5)}^{2}}$

Step 1.2

For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place $A$ .

$\frac{A}{{(x + 5)}^{2}}$

Step 1.3

$\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5}$

Step 1.4

$\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5} + \frac{C}{{(x - 5)}^{2}}$

Step 1.5

$\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5} + \frac{C}{{(x - 5)}^{2}} + \frac{D}{x - 5}$

Step 1.6

Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is ${(x + 5)}^{2} {(x - 5)}^{2}$ .

$\frac{5 {(x + 5)}^{2} {(x - 5)}^{2}}{{(x + 5)}^{2} {(x - 5)}^{2}} = \frac{(A) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.7

Cancel the common factor of ${(x + 5)}^{2}$ .

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Step 1.7.1

Cancel the common factor.

Step 1.7.2

Rewrite the expression.

$\frac{5 {(x - 5)}^{2}}{{(x - 5)}^{2}} = \frac{(A) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.8

Cancel the common factor of ${(x - 5)}^{2}$ .

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Step 1.8.1

Cancel the common factor.

Step 1.8.2

Divide $5$ by $1$ .

$5 = \frac{(A) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9

Simplify each term.

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Step 1.9.1

Cancel the common factor of ${(x + 5)}^{2}$ .

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Step 1.9.1.1

Cancel the common factor.

$5 = \frac{A {(x + 5)}^{2} {(x - 5)}^{2}}{{(x + 5)}^{2}} + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.1.2

Divide $(A) {(x - 5)}^{2}$ by $1$ .

$5 = (A) {(x - 5)}^{2} + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.2

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$5 = A ((x - 5) (x - 5)) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.3

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.9.3.1

Apply the distributive property.

$5 = A (x (x - 5) - 5 (x - 5)) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.3.2

Apply the distributive property.

$5 = A (x \cdot x + x \cdot - 5 - 5 (x - 5)) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.3.3

Apply the distributive property.

$5 = A (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.4

Simplify and combine like terms.

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Step 1.9.4.1

Simplify each term.

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Step 1.9.4.1.1

Multiply $x$ by $x$ .

$5 = A (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.4.1.2

Move $- 5$ to the left of $x$ .

$5 = A (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.4.1.3

Multiply $- 5$ by $- 5$ .

$5 = A (x^{2} - 5 x - 5 x + 25) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.4.2

Subtract $5 x$ from $- 5 x$ .

$5 = A (x^{2} - 10 x + 25) + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.5

Apply the distributive property.

$5 = A x^{2} + A (- 10 x) + A \cdot 25 + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.6

Simplify.

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Step 1.9.6.1

Rewrite using the commutative property of multiplication.

$5 = A x^{2} - 10 A x + A \cdot 25 + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.6.2

Move $25$ to the left of $A$ .

$5 = A x^{2} - 10 A x + 25 \cdot A + \frac{(B) {(x + 5)}^{2} {(x - 5)}^{2}}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.7

Cancel the common factor of ${(x + 5)}^{2}$ and $x + 5$ .

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Step 1.9.7.1

Factor $x + 5$ out of $(B) {(x + 5)}^{2} {(x - 5)}^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + \frac{(x + 5) (B (x + 5) {(x - 5)}^{2})}{x + 5} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.7.2

Cancel the common factors.

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Step 1.9.7.2.1

Multiply by $1$ .

$5 = A x^{2} - 10 A x + 25 A + \frac{(x + 5) (B (x + 5) {(x - 5)}^{2})}{(x + 5) \cdot 1} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.7.2.2

Cancel the common factor.

$5 = A x^{2} - 10 A x + 25 A + \frac{(x + 5) (B (x + 5) {(x - 5)}^{2})}{(x + 5) \cdot 1} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.7.2.3

Rewrite the expression.

$5 = A x^{2} - 10 A x + 25 A + \frac{B (x + 5) {(x - 5)}^{2}}{1} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.7.2.4

Divide $B (x + 5) {(x - 5)}^{2}$ by $1$ .

$5 = A x^{2} - 10 A x + 25 A + B (x + 5) {(x - 5)}^{2} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.8

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + (B x + B \cdot 5) {(x - 5)}^{2} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.9

Move $5$ to the left of $B$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 \cdot B) {(x - 5)}^{2} + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.10

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) ((x - 5) (x - 5)) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.11

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.9.11.1

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x (x - 5) - 5 (x - 5)) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.11.2

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x \cdot x + x \cdot - 5 - 5 (x - 5)) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.11.3

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.12

Simplify and combine like terms.

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Step 1.9.12.1

Simplify each term.

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Step 1.9.12.1.1

Multiply $x$ by $x$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.12.1.2

Move $- 5$ to the left of $x$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.12.1.3

Multiply $- 5$ by $- 5$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x^{2} - 5 x - 5 x + 25) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.12.2

Subtract $5 x$ from $- 5 x$ .

$5 = A x^{2} - 10 A x + 25 A + (B x + 5 B) (x^{2} - 10 x + 25) + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.13

Expand $(B x + 5 B) (x^{2} - 10 x + 25)$ by multiplying each term in the first expression by each term in the second expression.

$5 = A x^{2} - 10 A x + 25 A + B x \cdot x^{2} + B x (- 10 x) + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14

Simplify each term.

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Step 1.9.14.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.9.14.1.1

Move $x^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + B (x^{2} x) + B x (- 10 x) + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.9.14.1.2.1

Raise $x$ to the power of $1$ .

Step 1.9.14.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 = A x^{2} - 10 A x + 25 A + B x^{2 + 1} + B x (- 10 x) + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.1.3

Add $2$ and $1$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} + B x (- 10 x) + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.2

Rewrite using the commutative property of multiplication.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x \cdot x + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.14.3.1

Move $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B (x \cdot x) + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.3.2

Multiply $x$ by $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x^{2} + B x \cdot 25 + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.4

Move $25$ to the left of $B x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x^{2} + 25 \cdot (B x) + 5 B x^{2} + 5 B (- 10 x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.5

Rewrite using the commutative property of multiplication.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x^{2} + 25 B x + 5 B x^{2} + 5 \cdot (- 10 B x) + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.6

Multiply $5$ by $- 10$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x^{2} + 25 B x + 5 B x^{2} - 50 B x + 5 B \cdot 25 + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.14.7

Multiply $25$ by $5$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 10 B x^{2} + 25 B x + 5 B x^{2} - 50 B x + 125 B + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.15

Add $- 10 B x^{2}$ and $5 B x^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} + 25 B x - 50 B x + 125 B + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.16

Subtract $50 B x$ from $25 B x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.17

Cancel the common factor of ${(x - 5)}^{2}$ .

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Step 1.9.17.1

Cancel the common factor.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + \frac{(C) {(x + 5)}^{2} {(x - 5)}^{2}}{{(x - 5)}^{2}} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.17.2

Divide $(C) {(x + 5)}^{2}$ by $1$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + (C) {(x + 5)}^{2} + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.18

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C ((x + 5) (x + 5)) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.19

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.9.19.1

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x (x + 5) + 5 (x + 5)) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.19.2

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x \cdot x + x \cdot 5 + 5 (x + 5)) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.19.3

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.20

Simplify and combine like terms.

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Step 1.9.20.1

Simplify each term.

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Step 1.9.20.1.1

Multiply $x$ by $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.20.1.2

Move $5$ to the left of $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.20.1.3

Multiply $5$ by $5$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x^{2} + 5 x + 5 x + 25) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.20.2

Add $5 x$ and $5 x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C (x^{2} + 10 x + 25) + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.21

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + C (10 x) + C \cdot 25 + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.22

Simplify.

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Step 1.9.22.1

Rewrite using the commutative property of multiplication.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + C \cdot 25 + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.22.2

Move $25$ to the left of $C$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 \cdot C + \frac{(D) {(x + 5)}^{2} {(x - 5)}^{2}}{x - 5}$

Step 1.9.23

Cancel the common factor of ${(x - 5)}^{2}$ and $x - 5$ .

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Step 1.9.23.1

Factor $x - 5$ out of $(D) {(x + 5)}^{2} {(x - 5)}^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + \frac{(x - 5) ((D) {(x + 5)}^{2} (x - 5))}{x - 5}$

Step 1.9.23.2

Cancel the common factors.

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Step 1.9.23.2.1

Multiply by $1$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + \frac{(x - 5) ((D) {(x + 5)}^{2} (x - 5))}{(x - 5) \cdot 1}$

Step 1.9.23.2.2

Cancel the common factor.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + \frac{(x - 5) ((D) {(x + 5)}^{2} (x - 5))}{(x - 5) \cdot 1}$

Step 1.9.23.2.3

Rewrite the expression.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + \frac{(D) {(x + 5)}^{2} (x - 5)}{1}$

Step 1.9.23.2.4

Divide $(D) {(x + 5)}^{2} (x - 5)$ by $1$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + (D) {(x + 5)}^{2} (x - 5)$

Step 1.9.24

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D ((x + 5) (x + 5)) (x - 5)$

Step 1.9.25

Expand $(x + 5) (x + 5)$ using the FOIL Method.

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Step 1.9.25.1

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x (x + 5) + 5 (x + 5)) (x - 5)$

Step 1.9.25.2

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x \cdot x + x \cdot 5 + 5 (x + 5)) (x - 5)$

Step 1.9.25.3

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) (x - 5)$

Step 1.9.26

Simplify and combine like terms.

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Step 1.9.26.1

Simplify each term.

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Step 1.9.26.1.1

Multiply $x$ by $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) (x - 5)$

Step 1.9.26.1.2

Move $5$ to the left of $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) (x - 5)$

Step 1.9.26.1.3

Multiply $5$ by $5$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x^{2} + 5 x + 5 x + 25) (x - 5)$

Step 1.9.26.2

Add $5 x$ and $5 x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x^{2} + 10 x + 25) (x - 5)$

Step 1.9.27

Apply the distributive property.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + (D x^{2} + D (10 x) + D \cdot 25) (x - 5)$

Step 1.9.28

Simplify.

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Step 1.9.28.1

Rewrite using the commutative property of multiplication.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + (D x^{2} + 10 D x + D \cdot 25) (x - 5)$

Step 1.9.28.2

Move $25$ to the left of $D$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + (D x^{2} + 10 D x + 25 \cdot D) (x - 5)$

Step 1.9.29

Expand $(D x^{2} + 10 D x + 25 D) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{2} x + D x^{2} \cdot - 5 + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30

Simplify each term.

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Step 1.9.30.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.9.30.1.1

Move $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x \cdot x^{2}) + D x^{2} \cdot - 5 + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.9.30.1.2.1

Raise $x$ to the power of $1$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D (x \cdot x^{2}) + D x^{2} \cdot - 5 + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{1 + 2} + D x^{2} \cdot - 5 + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.1.3

Add $1$ and $2$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + D x^{2} \cdot - 5 + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.2

Move $- 5$ to the left of $D x^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} - 5 \cdot (D x^{2}) + 10 D x \cdot x + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.3

Multiply $x$ by $x$ by adding the exponents.

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Step 1.9.30.3.1

Move $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} - 5 D x^{2} + 10 D (x \cdot x) + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.3.2

Multiply $x$ by $x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} - 5 D x^{2} + 10 D x^{2} + 10 D x \cdot - 5 + 25 D x + 25 D \cdot - 5$

Step 1.9.30.4

Multiply $- 5$ by $10$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} - 5 D x^{2} + 10 D x^{2} - 50 D x + 25 D x + 25 D \cdot - 5$

Step 1.9.30.5

Multiply $- 5$ by $25$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} - 5 D x^{2} + 10 D x^{2} - 50 D x + 25 D x - 125 D$

Step 1.9.31

Add $- 5 D x^{2}$ and $10 D x^{2}$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 50 D x + 25 D x - 125 D$

Step 1.9.32

Add $- 50 D x$ and $25 D x$ .

$5 = A x^{2} - 10 A x + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 25 D x - 125 D$

Step 1.10

Simplify the expression.

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Step 1.10.1

Move $A$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 25 D x - 125 D$

Step 1.10.2

Reorder $B$ and $x^{3}$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 B x^{2} - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 25 D x - 125 D$

Step 1.10.3

Move $B$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 x^{2} B - 25 B x + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 25 D x - 125 D$

Step 1.10.4

Move $B$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 x^{2} B - 25 x B + 125 B + C x^{2} + 10 C x + 25 C + D x^{3} + 5 D x^{2} - 25 D x - 125 D$

Step 1.10.5

Move $25 C$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 x^{2} B - 25 x B + 125 B + C x^{2} + 10 C x + D x^{3} + 5 D x^{2} - 25 D x + 25 C - 125 D$

Step 1.10.6

Move $125 B$ .

$5 = A x^{2} - 10 x A + 25 A + B x^{3} - 5 x^{2} B - 25 x B + C x^{2} + 10 C x + D x^{3} + 5 D x^{2} - 25 D x + 125 B + 25 C - 125 D$

Step 1.10.7

Move $25 A$ .

$5 = A x^{2} - 10 x A + B x^{3} - 5 x^{2} B - 25 x B + C x^{2} + 10 C x + D x^{3} + 5 D x^{2} - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.8

Move $10 C x$ .

$5 = A x^{2} - 10 x A + B x^{3} - 5 x^{2} B - 25 x B + C x^{2} + D x^{3} + 5 D x^{2} + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.9

Move $- 25 x B$ .

$5 = A x^{2} - 10 x A + B x^{3} - 5 x^{2} B + C x^{2} + D x^{3} + 5 D x^{2} - 25 x B + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.10

Move $- 10 x A$ .

$5 = A x^{2} + B x^{3} - 5 x^{2} B + C x^{2} + D x^{3} + 5 D x^{2} - 10 x A - 25 x B + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.11

Move $C x^{2}$ .

$5 = A x^{2} + B x^{3} - 5 x^{2} B + D x^{3} + C x^{2} + 5 D x^{2} - 10 x A - 25 x B + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.12

Move $- 5 x^{2} B$ .

$5 = A x^{2} + B x^{3} + D x^{3} - 5 x^{2} B + C x^{2} + 5 D x^{2} - 10 x A - 25 x B + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 1.10.13

Move $A x^{2}$ .

$5 = B x^{3} + D x^{3} + A x^{2} - 5 x^{2} B + C x^{2} + 5 D x^{2} - 10 x A - 25 x B + 10 C x - 25 D x + 25 A + 125 B + 25 C - 125 D$

Step 2

Create equations for the partial fraction variables and use them to set up a system of equations.

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Step 2.1

Create an equation for the partial fraction variables by equating the coefficients of $x^{3}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = B + D$

Step 2.2

Create an equation for the partial fraction variables by equating the coefficients of $x^{2}$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = A - 5 B + C + 5 D$

Step 2.3

Create an equation for the partial fraction variables by equating the coefficients of $x$ from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$0 = - 10 A - 25 B + 10 C - 25 D$

Step 2.4

Create an equation for the partial fraction variables by equating the coefficients of the terms not containing $x$ . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.

$5 = 25 A + 125 B + 25 C - 125 D$

Step 2.5

Set up the system of equations to find the coefficients of the partial fractions.

$0 = B + D$

$0 = A - 5 B + C + 5 D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = B + D$

$0 = A - 5 B + C + 5 D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3

Solve the system of equations.

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Step 3.1

Solve for $B$ in $0 = B + D$ .

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Step 3.1.1

Rewrite the equation as $B + D = 0$ .

$B + D = 0$

$0 = A - 5 B + C + 5 D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.1.2

Subtract $D$ from both sides of the equation.

$B = - D$

$0 = A - 5 B + C + 5 D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

$B = - D$

$0 = A - 5 B + C + 5 D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2

Replace all occurrences of $B$ with $- D$ in each equation.

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Step 3.2.1

Replace all occurrences of $B$ in $0 = A - 5 B + C + 5 D$ with $- D$ .

$0 = A - 5 (- D) + C + 5 D$

$B = - D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $A - 5 (- D) + C + 5 D$ .

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Step 3.2.2.1.1

Multiply $- 1$ by $- 5$ .

$0 = A + 5 D + C + 5 D$

$B = - D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.2.1.2

Add $5 D$ and $5 D$ .

$0 = A + C + 10 D$

$B = - D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = A + C + 10 D$

$B = - D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = A + C + 10 D$

$B = - D$

$0 = - 10 A - 25 B + 10 C - 25 D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.3

Replace all occurrences of $B$ in $0 = - 10 A - 25 B + 10 C - 25 D$ with $- D$ .

$0 = - 10 A - 25 (- D) + 10 C - 25 D$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $- 10 A - 25 (- D) + 10 C - 25 D$ .

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Step 3.2.4.1.1

Multiply $- 1$ by $- 25$ .

$0 = - 10 A + 25 D + 10 C - 25 D$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.4.1.2

Combine the opposite terms in $- 10 A + 25 D + 10 C - 25 D$ .

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Step 3.2.4.1.2.1

Subtract $25 D$ from $25 D$ .

$0 = - 10 A + 10 C + 0$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.4.1.2.2

Add $- 10 A + 10 C$ and $0$ .

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 125 B + 25 C - 125 D$

Step 3.2.5

Replace all occurrences of $B$ in $5 = 25 A + 125 B + 25 C - 125 D$ with $- D$ .

$5 = 25 A + 125 (- D) + 25 C - 125 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

Step 3.2.6

Simplify the right side.

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Step 3.2.6.1

Simplify $25 A + 125 (- D) + 25 C - 125 D$ .

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Step 3.2.6.1.1

Multiply $- 1$ by $125$ .

$5 = 25 A - 125 D + 25 C - 125 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

Step 3.2.6.1.2

Subtract $125 D$ from $- 125 D$ .

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$0 = A + C + 10 D$

$B = - D$

Step 3.3

Solve for $A$ in $0 = A + C + 10 D$ .

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Step 3.3.1

Rewrite the equation as $A + C + 10 D = 0$ .

$A + C + 10 D = 0$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.3.2

Move all terms not containing $A$ to the right side of the equation.

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Step 3.3.2.1

Subtract $C$ from both sides of the equation.

$A + 10 D = - C$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.3.2.2

Subtract $10 D$ from both sides of the equation.

$A = - C - 10 D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$B = - D$

$A = - C - 10 D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$B = - D$

$A = - C - 10 D$

$5 = 25 A + 25 C - 250 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4

Replace all occurrences of $A$ with $- C - 10 D$ in each equation.

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Step 3.4.1

Replace all occurrences of $A$ in $5 = 25 A + 25 C - 250 D$ with $- C - 10 D$ .

$5 = 25 (- C - 10 D) + 25 C - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $25 (- C - 10 D) + 25 C - 250 D$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$5 = 25 (- C) + 25 (- 10 D) + 25 C - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2.1.1.2

Multiply $- 1$ by $25$ .

$5 = - 25 C + 25 (- 10 D) + 25 C - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2.1.1.3

Multiply $- 10$ by $25$ .

$5 = - 25 C - 250 D + 25 C - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

$5 = - 25 C - 250 D + 25 C - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2.1.2

Simplify by adding terms.

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Step 3.4.2.1.2.1

Combine the opposite terms in $- 25 C - 250 D + 25 C - 250 D$ .

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Step 3.4.2.1.2.1.1

Add $- 25 C$ and $25 C$ .

$5 = - 250 D + 0 - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2.1.2.1.2

Add $- 250 D$ and $0$ .

$5 = - 250 D - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

$5 = - 250 D - 250 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.2.1.2.2

Subtract $250 D$ from $- 250 D$ .

$5 = - 500 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

$5 = - 500 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

$5 = - 500 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

$5 = - 500 D$

$A = - C - 10 D$

$0 = - 10 A + 10 C$

$B = - D$

Step 3.4.3

Replace all occurrences of $A$ in $0 = - 10 A + 10 C$ with $- C - 10 D$ .

$0 = - 10 (- C - 10 D) + 10 C$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $- 10 (- C - 10 D) + 10 C$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$0 = - 10 (- C) - 10 (- 10 D) + 10 C$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

Step 3.4.4.1.1.2

Multiply $- 1$ by $- 10$ .

$0 = 10 C - 10 (- 10 D) + 10 C$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

Step 3.4.4.1.1.3

Multiply $- 10$ by $- 10$ .

$0 = 10 C + 100 D + 10 C$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

$0 = 10 C + 100 D + 10 C$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

Step 3.4.4.1.2

Add $10 C$ and $10 C$ .

$0 = 20 C + 100 D$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

$0 = 20 C + 100 D$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

$0 = 20 C + 100 D$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

$0 = 20 C + 100 D$

$5 = - 500 D$

$A = - C - 10 D$

$B = - D$

Step 3.5

Solve for $D$ in $5 = - 500 D$ .

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Step 3.5.1

Rewrite the equation as $- 500 D = 5$ .

$- 500 D = 5$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2

Divide each term in $- 500 D = 5$ by $- 500$ and simplify.

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Step 3.5.2.1

Divide each term in $- 500 D = 5$ by $- 500$ .

$\frac{- 500 D}{- 500} = \frac{5}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.2

Simplify the left side.

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Step 3.5.2.2.1

Cancel the common factor of $- 500$ .

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Step 3.5.2.2.1.1

Cancel the common factor.

$\frac{- 500 D}{- 500} = \frac{5}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.2.1.2

Divide $D$ by $1$ .

$D = \frac{5}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = \frac{5}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = \frac{5}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.3

Simplify the right side.

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Step 3.5.2.3.1

Cancel the common factor of $5$ and $- 500$ .

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Step 3.5.2.3.1.1

Factor $5$ out of $5$ .

$D = \frac{5 (1)}{- 500}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.3.1.2

Cancel the common factors.

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Step 3.5.2.3.1.2.1

Factor $5$ out of $- 500$ .

$D = \frac{5 \cdot 1}{5 \cdot - 100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.3.1.2.2

Cancel the common factor.

$D = \frac{5 \cdot 1}{5 \cdot - 100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.3.1.2.3

Rewrite the expression.

$D = \frac{1}{- 100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = \frac{1}{- 100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = \frac{1}{- 100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.5.2.3.2

Move the negative in front of the fraction.

$D = - \frac{1}{100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = - \frac{1}{100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = - \frac{1}{100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

$D = - \frac{1}{100}$

$0 = 20 C + 100 D$

$A = - C - 10 D$

$B = - D$

Step 3.6

Replace all occurrences of $D$ with $- \frac{1}{100}$ in each equation.

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Step 3.6.1

Replace all occurrences of $D$ in $0 = 20 C + 100 D$ with $- \frac{1}{100}$ .

$0 = 20 C + 100 (- \frac{1}{100})$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Cancel the common factor of $100$ .

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Step 3.6.2.1.1

Move the leading negative in $- \frac{1}{100}$ into the numerator.

$0 = 20 C + 100 (\frac{- 1}{100})$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

Step 3.6.2.1.2

Cancel the common factor.

$0 = 20 C + 100 (\frac{- 1}{100})$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

Step 3.6.2.1.3

Rewrite the expression.

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$A = - C - 10 D$

$B = - D$

Step 3.6.3

Replace all occurrences of $D$ in $A = - C - 10 D$ with $- \frac{1}{100}$ .

$A = - C - 10 (- \frac{1}{100})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify each term.

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Step 3.6.4.1.1

Cancel the common factor of $10$ .

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Step 3.6.4.1.1.1

Move the leading negative in $- \frac{1}{100}$ into the numerator.

$A = - C - 10 (\frac{- 1}{100})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.1.2

Factor $10$ out of $- 10$ .

$A = - C + 10 (- 1) (\frac{- 1}{100})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.1.3

Factor $10$ out of $100$ .

$A = - C + 10 \cdot (- 1 \frac{- 1}{10 \cdot 10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.1.4

Cancel the common factor.

$A = - C + 10 \cdot (- 1 \frac{- 1}{10 \cdot 10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.1.5

Rewrite the expression.

$A = - C - 1 (\frac{- 1}{10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

$A = - C - 1 (\frac{- 1}{10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.2

Move the negative in front of the fraction.

$A = - C - 1 (- \frac{1}{10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.3

Multiply $- 1 (- \frac{1}{10})$ .

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Step 3.6.4.1.3.1

Multiply $- 1$ by $- 1$ .

$A = - C + 1 (\frac{1}{10})$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.4.1.3.2

Multiply $\frac{1}{10}$ by $1$ .

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = - D$

Step 3.6.5

Replace all occurrences of $D$ in $B = - D$ with $- \frac{1}{100}$ .

$B = - (- \frac{1}{100})$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

Step 3.6.6

Simplify the right side.

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Step 3.6.6.1

Multiply $- (- \frac{1}{100})$ .

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Step 3.6.6.1.1

Multiply $- 1$ by $- 1$ .

$B = 1 (\frac{1}{100})$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

Step 3.6.6.1.2

Multiply $\frac{1}{100}$ by $1$ .

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$0 = 20 C - 1$

$D = - \frac{1}{100}$

Step 3.7

Solve for $C$ in $0 = 20 C - 1$ .

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Step 3.7.1

Rewrite the equation as $20 C - 1 = 0$ .

$20 C - 1 = 0$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

Step 3.7.2

Add $1$ to both sides of the equation.

$20 C = 1$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

Step 3.7.3

Divide each term in $20 C = 1$ by $20$ and simplify.

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Step 3.7.3.1

Divide each term in $20 C = 1$ by $20$ .

$\frac{20 C}{20} = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

Step 3.7.3.2

Simplify the left side.

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Step 3.7.3.2.1

Cancel the common factor of $20$ .

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Step 3.7.3.2.1.1

Cancel the common factor.

$\frac{20 C}{20} = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

Step 3.7.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$A = - C + \frac{1}{10}$

$D = - \frac{1}{100}$

Step 3.8

Replace all occurrences of $C$ with $\frac{1}{20}$ in each equation.

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Step 3.8.1

Replace all occurrences of $C$ in $A = - C + \frac{1}{10}$ with $\frac{1}{20}$ .

$A = - (\frac{1}{20}) + \frac{1}{10}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

Simplify $- (\frac{1}{20}) + \frac{1}{10}$ .

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Step 3.8.2.1.1

To write $\frac{1}{10}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A = - \frac{1}{20} + \frac{1}{10} \cdot \frac{2}{2}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.8.2.1.2

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.2.1.2.1

Multiply $\frac{1}{10}$ by $\frac{2}{2}$ .

$A = - \frac{1}{20} + \frac{2}{10 \cdot 2}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.8.2.1.2.2

Multiply $10$ by $2$ .

$A = - \frac{1}{20} + \frac{2}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

$A = - \frac{1}{20} + \frac{2}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.8.2.1.3

Combine the numerators over the common denominator.

$A = \frac{- 1 + 2}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.8.2.1.4

Add $- 1$ and $2$ .

$A = \frac{1}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

$A = \frac{1}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

$A = \frac{1}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

$A = \frac{1}{20}$

$C = \frac{1}{20}$

$B = \frac{1}{100}$

$D = - \frac{1}{100}$

Step 3.9

List all of the solutions.

$A = \frac{1}{20}, C = \frac{1}{20}, B = \frac{1}{100}, D = - \frac{1}{100}$

Step 4

Replace each of the partial fraction coefficients in $\frac{A}{{(x + 5)}^{2}} + \frac{B}{x + 5} + \frac{C}{{(x - 5)}^{2}} + \frac{D}{x - 5}$ with the values found for $A$ , $B$ , $C$ , and $D$ .

$\frac{\frac{1}{20}}{{(x + 5)}^{2}} + \frac{\frac{1}{100}}{x + 5} + \frac{\frac{1}{20}}{{(x - 5)}^{2}} + \frac{- \frac{1}{100}}{x - 5}$