Precalculus Examples

$\frac{7 x^{2}}{{(x - 5)}^{2} {(x + 5)}^{2}}$

Step 1

Decompose the fraction and multiply through by the common denominator.

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Step 1.1

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 1.2

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

Step 1.3

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

Step 1.4

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

Step 1.5

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

$\frac{7 x^{2} {(x - 5)}^{2} {(x + 5)}^{2}}{{(x - 5)}^{2} {(x + 5)}^{2}} = \frac{(A) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x - 5)}^{2}} + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.6

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

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Step 1.6.1

Cancel the common factor.

Step 1.6.2

Rewrite the expression.

$\frac{7 x^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} = \frac{(A) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x - 5)}^{2}} + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

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

Divide $7 x^{2}$ by $1$ .

$7 x^{2} = \frac{(A) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x - 5)}^{2}} + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8

Simplify each term.

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Step 1.8.1

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

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Step 1.8.1.1

Cancel the common factor.

$7 x^{2} = \frac{A {(x - 5)}^{2} {(x + 5)}^{2}}{{(x - 5)}^{2}} + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.1.2

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

$7 x^{2} = (A) {(x + 5)}^{2} + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.2

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

$7 x^{2} = A ((x + 5) (x + 5)) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.3

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

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Step 1.8.3.1

Apply the distributive property.

$7 x^{2} = A (x (x + 5) + 5 (x + 5)) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.3.2

Apply the distributive property.

$7 x^{2} = A (x \cdot x + x \cdot 5 + 5 (x + 5)) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.3.3

Apply the distributive property.

$7 x^{2} = A (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.4

Simplify and combine like terms.

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Step 1.8.4.1

Simplify each term.

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Step 1.8.4.1.1

Multiply $x$ by $x$ .

$7 x^{2} = A (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.4.1.2

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

$7 x^{2} = A (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.4.1.3

Multiply $5$ by $5$ .

$7 x^{2} = A (x^{2} + 5 x + 5 x + 25) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.4.2

Add $5 x$ and $5 x$ .

$7 x^{2} = A (x^{2} + 10 x + 25) + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.5

Apply the distributive property.

$7 x^{2} = A x^{2} + A (10 x) + A \cdot 25 + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.6

Simplify.

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Step 1.8.6.1

Rewrite using the commutative property of multiplication.

$7 x^{2} = A x^{2} + 10 A x + A \cdot 25 + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.6.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 \cdot A + \frac{(B) {(x - 5)}^{2} {(x + 5)}^{2}}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.7

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

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Step 1.8.7.1

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + \frac{(x - 5) (B (x - 5) {(x + 5)}^{2})}{x - 5} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.7.2

Cancel the common factors.

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Step 1.8.7.2.1

Multiply by $1$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + \frac{(x - 5) (B (x - 5) {(x + 5)}^{2})}{(x - 5) \cdot 1} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.7.2.2

Cancel the common factor.

$7 x^{2} = A x^{2} + 10 A x + 25 A + \frac{(x - 5) (B (x - 5) {(x + 5)}^{2})}{(x - 5) \cdot 1} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.7.2.3

Rewrite the expression.

$7 x^{2} = A x^{2} + 10 A x + 25 A + \frac{B (x - 5) {(x + 5)}^{2}}{1} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.7.2.4

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B (x - 5) {(x + 5)}^{2} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.8

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x + B \cdot - 5) {(x + 5)}^{2} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.9

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 \cdot B) {(x + 5)}^{2} + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.10

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) ((x + 5) (x + 5)) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.11

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

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Step 1.8.11.1

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x (x + 5) + 5 (x + 5)) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.11.2

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x \cdot x + x \cdot 5 + 5 (x + 5)) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.11.3

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.12

Simplify and combine like terms.

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Step 1.8.12.1

Simplify each term.

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Step 1.8.12.1.1

Multiply $x$ by $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.12.1.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.12.1.3

Multiply $5$ by $5$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x^{2} + 5 x + 5 x + 25) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.12.2

Add $5 x$ and $5 x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + (B x - 5 B) (x^{2} + 10 x + 25) + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.13

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.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x \cdot x^{2} + B x (10 x) + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14

Simplify each term.

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Step 1.8.14.1

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

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Step 1.8.14.1.1

Move $x^{2}$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B (x^{2} x) + B x (10 x) + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.1.2

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

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Step 1.8.14.1.2.1

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

Step 1.8.14.1.2.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{2 + 1} + B x (10 x) + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.1.3

Add $2$ and $1$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + B x (10 x) + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.2

Rewrite using the commutative property of multiplication.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x \cdot x + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.3

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

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Step 1.8.14.3.1

Move $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B (x \cdot x) + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.3.2

Multiply $x$ by $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x^{2} + B x \cdot 25 - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.4

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x^{2} + 25 \cdot (B x) - 5 B x^{2} - 5 B (10 x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.5

Rewrite using the commutative property of multiplication.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x^{2} + 25 B x - 5 B x^{2} - 5 \cdot (10 B x) - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.6

Multiply $- 5$ by $10$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x^{2} + 25 B x - 5 B x^{2} - 50 B x - 5 B \cdot 25 + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.14.7

Multiply $25$ by $- 5$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 10 B x^{2} + 25 B x - 5 B x^{2} - 50 B x - 125 B + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.15

Subtract $5 B x^{2}$ from $10 B x^{2}$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} + 25 B x - 50 B x - 125 B + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.16

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.17

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

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Step 1.8.17.1

Cancel the common factor.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + \frac{(C) {(x - 5)}^{2} {(x + 5)}^{2}}{{(x + 5)}^{2}} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.17.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + (C) {(x - 5)}^{2} + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.18

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C ((x - 5) (x - 5)) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.19

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

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Step 1.8.19.1

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x (x - 5) - 5 (x - 5)) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.19.2

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x \cdot x + x \cdot - 5 - 5 (x - 5)) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.19.3

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.20

Simplify and combine like terms.

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Step 1.8.20.1

Simplify each term.

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Step 1.8.20.1.1

Multiply $x$ by $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.20.1.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.20.1.3

Multiply $- 5$ by $- 5$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x^{2} - 5 x - 5 x + 25) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.20.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C (x^{2} - 10 x + 25) + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.21

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} + C (- 10 x) + C \cdot 25 + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.22

Simplify.

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Step 1.8.22.1

Rewrite using the commutative property of multiplication.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + C \cdot 25 + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.22.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 \cdot C + \frac{(D) {(x - 5)}^{2} {(x + 5)}^{2}}{x + 5}$

Step 1.8.23

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

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Step 1.8.23.1

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + \frac{(x + 5) ((D) {(x - 5)}^{2} (x + 5))}{x + 5}$

Step 1.8.23.2

Cancel the common factors.

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Step 1.8.23.2.1

Multiply by $1$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + \frac{(x + 5) ((D) {(x - 5)}^{2} (x + 5))}{(x + 5) \cdot 1}$

Step 1.8.23.2.2

Cancel the common factor.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + \frac{(x + 5) ((D) {(x - 5)}^{2} (x + 5))}{(x + 5) \cdot 1}$

Step 1.8.23.2.3

Rewrite the expression.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + \frac{(D) {(x - 5)}^{2} (x + 5)}{1}$

Step 1.8.23.2.4

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + (D) {(x - 5)}^{2} (x + 5)$

Step 1.8.24

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D ((x - 5) (x - 5)) (x + 5)$

Step 1.8.25

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

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Step 1.8.25.1

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x (x - 5) - 5 (x - 5)) (x + 5)$

Step 1.8.25.2

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x \cdot x + x \cdot - 5 - 5 (x - 5)) (x + 5)$

Step 1.8.25.3

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) (x + 5)$

Step 1.8.26

Simplify and combine like terms.

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Step 1.8.26.1

Simplify each term.

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Step 1.8.26.1.1

Multiply $x$ by $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) (x + 5)$

Step 1.8.26.1.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) (x + 5)$

Step 1.8.26.1.3

Multiply $- 5$ by $- 5$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x^{2} - 5 x - 5 x + 25) (x + 5)$

Step 1.8.26.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x^{2} - 10 x + 25) (x + 5)$

Step 1.8.27

Apply the distributive property.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + (D x^{2} + D (- 10 x) + D \cdot 25) (x + 5)$

Step 1.8.28

Simplify.

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Step 1.8.28.1

Rewrite using the commutative property of multiplication.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + (D x^{2} - 10 D x + D \cdot 25) (x + 5)$

Step 1.8.28.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + (D x^{2} - 10 D x + 25 \cdot D) (x + 5)$

Step 1.8.29

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.

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{2} x + D x^{2} \cdot 5 - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30

Simplify each term.

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Step 1.8.30.1

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

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Step 1.8.30.1.1

Move $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x \cdot x^{2}) + D x^{2} \cdot 5 - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.1.2

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

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Step 1.8.30.1.2.1

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D (x \cdot x^{2}) + D x^{2} \cdot 5 - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.1.2.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{1 + 2} + D x^{2} \cdot 5 - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.1.3

Add $1$ and $2$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + D x^{2} \cdot 5 - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.2

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + 5 \cdot (D x^{2}) - 10 D x \cdot x - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.3

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

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Step 1.8.30.3.1

Move $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + 5 D x^{2} - 10 D (x \cdot x) - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.3.2

Multiply $x$ by $x$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + 5 D x^{2} - 10 D x^{2} - 10 D x \cdot 5 + 25 D x + 25 D \cdot 5$

Step 1.8.30.4

Multiply $5$ by $- 10$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + 5 D x^{2} - 10 D x^{2} - 50 D x + 25 D x + 25 D \cdot 5$

Step 1.8.30.5

Multiply $5$ by $25$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} + 5 D x^{2} - 10 D x^{2} - 50 D x + 25 D x + 125 D$

Step 1.8.31

Subtract $10 D x^{2}$ from $5 D x^{2}$ .

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 50 D x + 25 D x + 125 D$

Step 1.8.32

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

$7 x^{2} = A x^{2} + 10 A x + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 25 D x + 125 D$

Step 1.9

Simplify the expression.

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Step 1.9.1

Move $A$ .

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 25 D x + 125 D$

Step 1.9.2

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

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 B x^{2} - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 25 D x + 125 D$

Step 1.9.3

Move $B$ .

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 x^{2} B - 25 B x - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 25 D x + 125 D$

Step 1.9.4

Move $B$ .

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 x^{2} B - 25 x B - 125 B + C x^{2} - 10 C x + 25 C + D x^{3} - 5 D x^{2} - 25 D x + 125 D$

Step 1.9.5

Move $25 C$ .

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 x^{2} B - 25 x B - 125 B + C x^{2} - 10 C x + D x^{3} - 5 D x^{2} - 25 D x + 25 C + 125 D$

Step 1.9.6

Move $- 125 B$ .

$7 x^{2} = A x^{2} + 10 x A + 25 A + B x^{3} + 5 x^{2} B - 25 x B + C x^{2} - 10 C x + D x^{3} - 5 D x^{2} - 25 D x - 125 B + 25 C + 125 D$

Step 1.9.7

Move $25 A$ .

$7 x^{2} = A x^{2} + 10 x A + B x^{3} + 5 x^{2} B - 25 x B + C x^{2} - 10 C x + D x^{3} - 5 D x^{2} - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.8

Move $- 10 C x$ .

$7 x^{2} = A x^{2} + 10 x A + B x^{3} + 5 x^{2} B - 25 x B + C x^{2} + D x^{3} - 5 D x^{2} - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.9

Move $- 25 x B$ .

$7 x^{2} = A x^{2} + 10 x A + B x^{3} + 5 x^{2} B + C x^{2} + D x^{3} - 5 D x^{2} - 25 x B - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.10

Move $10 x A$ .

$7 x^{2} = A x^{2} + B x^{3} + 5 x^{2} B + C x^{2} + D x^{3} - 5 D x^{2} + 10 x A - 25 x B - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.11

Move $C x^{2}$ .

$7 x^{2} = A x^{2} + B x^{3} + 5 x^{2} B + D x^{3} + C x^{2} - 5 D x^{2} + 10 x A - 25 x B - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.12

Move $5 x^{2} B$ .

$7 x^{2} = A x^{2} + B x^{3} + D x^{3} + 5 x^{2} B + C x^{2} - 5 D x^{2} + 10 x A - 25 x B - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

Step 1.9.13

Move $A x^{2}$ .

$7 x^{2} = B x^{3} + D x^{3} + A x^{2} + 5 x^{2} B + C x^{2} - 5 D x^{2} + 10 x A - 25 x B - 10 C x - 25 D x + 25 A - 125 B + 25 C + 125 D$

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.

$7 = A + 5 B + C - 5 D$

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.

$0 = 25 A - 125 B + 25 C + 125 D$

Step 2.5

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

$0 = B + D$

$7 = A + 5 B + C - 5 D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

$0 = B + D$

$7 = A + 5 B + C - 5 D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

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 $B + D = 0$ .

$B + D = 0$

$7 = A + 5 B + C - 5 D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

Step 3.1.2

Subtract $D$ from both sides of the equation.

$B = - D$

$7 = A + 5 B + C - 5 D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

$B = - D$

$7 = A + 5 B + C - 5 D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

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 $7 = A + 5 B + C - 5 D$ with $- D$ .

$7 = A + 5 (- D) + C - 5 D$

$B = - D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

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 $5$ .

$7 = A - 5 D + C - 5 D$

$B = - D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

Step 3.2.2.1.2

Subtract $5 D$ from $- 5 D$ .

$7 = A + C - 10 D$

$B = - D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

$7 = A + C - 10 D$

$B = - D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

$7 = A + C - 10 D$

$B = - D$

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

$0 = 25 A - 125 B + 25 C + 125 D$

Step 3.2.3

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

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

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

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 $- 25$ .

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

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

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 $25 D$ .

$0 = 10 A - 10 C + 0$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

Step 3.2.4.1.2.2

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

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A - 125 B + 25 C + 125 D$

Step 3.2.5

Replace all occurrences of $B$ in $0 = 25 A - 125 B + 25 C + 125 D$ with $- D$ .

$0 = 25 A - 125 (- D) + 25 C + 125 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

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 $- 125$ .

$0 = 25 A + 125 D + 25 C + 125 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

Step 3.2.6.1.2

Add $125 D$ and $125 D$ .

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$7 = A + C - 10 D$

$B = - D$

Step 3.3

Solve for $A$ in $7 = A + C - 10 D$ .

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Step 3.3.1

Rewrite the equation as $A + C - 10 D = 7$ .

$A + C - 10 D = 7$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$B = - D$

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.

$A - 10 D = 7 - C$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.3.2.2

Add $10 D$ to both sides of the equation.

$A = 7 - C + 10 D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$B = - D$

$A = 7 - C + 10 D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$B = - D$

$A = 7 - C + 10 D$

$0 = 25 A + 25 C + 250 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4

Replace all occurrences of $A$ with $7 - C + 10 D$ in each equation.

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Step 3.4.1

Replace all occurrences of $A$ in $0 = 25 A + 25 C + 250 D$ with $7 - C + 10 D$ .

$0 = 25 (7 - C + 10 D) + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $25 (7 - C + 10 D) + 25 C + 250 D$ .

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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.

$0 = 25 \cdot 7 + 25 (- C) + 25 (10 D) + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2.1.1.2

Simplify.

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Step 3.4.2.1.1.2.1

Multiply $25$ by $7$ .

$0 = 175 + 25 (- C) + 25 (10 D) + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2.1.1.2.2

Multiply $- 1$ by $25$ .

$0 = 175 - 25 C + 25 (10 D) + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2.1.1.2.3

Multiply $10$ by $25$ .

$0 = 175 - 25 C + 250 D + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 - 25 C + 250 D + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 - 25 C + 250 D + 25 C + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

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 $175 - 25 C + 250 D + 25 C + 250 D$ .

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Step 3.4.2.1.2.1.1

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

$0 = 175 + 250 D + 0 + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2.1.2.1.2

Add $175 + 250 D$ and $0$ .

$0 = 175 + 250 D + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 + 250 D + 250 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.2.1.2.2

Add $250 D$ and $250 D$ .

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$0 = 10 A - 10 C$

$B = - D$

Step 3.4.3

Replace all occurrences of $A$ in $0 = 10 A - 10 C$ with $7 - C + 10 D$ .

$0 = 10 (7 - C + 10 D) - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $10 (7 - C + 10 D) - 10 C$ .

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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.

$0 = 10 \cdot 7 + 10 (- C) + 10 (10 D) - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.4.4.1.1.2

Simplify.

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Step 3.4.4.1.1.2.1

Multiply $10$ by $7$ .

$0 = 70 + 10 (- C) + 10 (10 D) - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.4.4.1.1.2.2

Multiply $- 1$ by $10$ .

$0 = 70 - 10 C + 10 (10 D) - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.4.4.1.1.2.3

Multiply $10$ by $10$ .

$0 = 70 - 10 C + 100 D - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 10 C + 100 D - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 10 C + 100 D - 10 C$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.4.4.1.2

Subtract $10 C$ from $- 10 C$ .

$0 = 70 - 20 C + 100 D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 20 C + 100 D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 20 C + 100 D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 20 C + 100 D$

$0 = 175 + 500 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5

Solve for $D$ in $0 = 175 + 500 D$ .

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Step 3.5.1

Rewrite the equation as $175 + 500 D = 0$ .

$175 + 500 D = 0$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.2

Subtract $175$ from both sides of the equation.

$500 D = - 175$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3

Divide each term in $500 D = - 175$ by $500$ and simplify.

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Step 3.5.3.1

Divide each term in $500 D = - 175$ by $500$ .

$\frac{500 D}{500} = \frac{- 175}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.2

Simplify the left side.

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Step 3.5.3.2.1

Cancel the common factor of $500$ .

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Step 3.5.3.2.1.1

Cancel the common factor.

$\frac{500 D}{500} = \frac{- 175}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.2.1.2

Divide $D$ by $1$ .

$D = \frac{- 175}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = \frac{- 175}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = \frac{- 175}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.3

Simplify the right side.

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Step 3.5.3.3.1

Cancel the common factor of $- 175$ and $500$ .

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Step 3.5.3.3.1.1

Factor $25$ out of $- 175$ .

$D = \frac{25 (- 7)}{500}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.3.1.2

Cancel the common factors.

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Step 3.5.3.3.1.2.1

Factor $25$ out of $500$ .

$D = \frac{25 \cdot - 7}{25 \cdot 20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.3.1.2.2

Cancel the common factor.

$D = \frac{25 \cdot - 7}{25 \cdot 20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.3.1.2.3

Rewrite the expression.

$D = \frac{- 7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = \frac{- 7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = \frac{- 7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.5.3.3.2

Move the negative in front of the fraction.

$D = - \frac{7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = - \frac{7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = - \frac{7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

$D = - \frac{7}{20}$

$0 = 70 - 20 C + 100 D$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6

Replace all occurrences of $D$ with $- \frac{7}{20}$ in each equation.

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Step 3.6.1

Replace all occurrences of $D$ in $0 = 70 - 20 C + 100 D$ with $- \frac{7}{20}$ .

$0 = 70 - 20 C + 100 (- \frac{7}{20})$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $70 - 20 C + 100 (- \frac{7}{20})$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Cancel the common factor of $20$ .

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Step 3.6.2.1.1.1.1

Move the leading negative in $- \frac{7}{20}$ into the numerator.

$0 = 70 - 20 C + 100 (\frac{- 7}{20})$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2.1.1.1.2

Factor $20$ out of $100$ .

$0 = 70 - 20 C + 20 (5) (\frac{- 7}{20})$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2.1.1.1.3

Cancel the common factor.

$0 = 70 - 20 C + 20 \cdot (5 (\frac{- 7}{20}))$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2.1.1.1.4

Rewrite the expression.

$0 = 70 - 20 C + 5 \cdot - 7$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 20 C + 5 \cdot - 7$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2.1.1.2

Multiply $5$ by $- 7$ .

$0 = 70 - 20 C - 35$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

$0 = 70 - 20 C - 35$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.2.1.2

Subtract $35$ from $70$ .

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$A = 7 - C + 10 D$

$B = - D$

Step 3.6.3

Replace all occurrences of $D$ in $A = 7 - C + 10 D$ with $- \frac{7}{20}$ .

$A = 7 - C + 10 (- \frac{7}{20})$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $7 - C + 10 (- \frac{7}{20})$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Cancel the common factor of $10$ .

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Step 3.6.4.1.1.1.1

Move the leading negative in $- \frac{7}{20}$ into the numerator.

$A = 7 - C + 10 (\frac{- 7}{20})$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.1.1.2

Factor $10$ out of $20$ .

$A = 7 - C + 10 (\frac{- 7}{10 (2)})$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.1.1.3

Cancel the common factor.

$A = 7 - C + 10 (\frac{- 7}{10 \cdot 2})$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.1.1.4

Rewrite the expression.

$A = 7 - C + \frac{- 7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

$A = 7 - C + \frac{- 7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.1.2

Move the negative in front of the fraction.

$A = 7 - C - \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

$A = 7 - C - \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.2

To write $7$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A = - C + 7 \cdot \frac{2}{2} - \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.3

Combine $7$ and $\frac{2}{2}$ .

$A = - C + \frac{7 \cdot 2}{2} - \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.4

Combine the numerators over the common denominator.

$A = - C + \frac{7 \cdot 2 - 7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.5

Simplify the numerator.

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Step 3.6.4.1.5.1

Multiply $7$ by $2$ .

$A = - C + \frac{14 - 7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.4.1.5.2

Subtract $7$ from $14$ .

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = - D$

Step 3.6.5

Replace all occurrences of $D$ in $B = - D$ with $- \frac{7}{20}$ .

$B = - (- \frac{7}{20})$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

Step 3.6.6

Simplify the right side.

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Step 3.6.6.1

Multiply $- (- \frac{7}{20})$ .

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Step 3.6.6.1.1

Multiply $- 1$ by $- 1$ .

$B = 1 (\frac{7}{20})$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

Step 3.6.6.1.2

Multiply $\frac{7}{20}$ by $1$ .

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$0 = - 20 C + 35$

$D = - \frac{7}{20}$

Step 3.7

Solve for $C$ in $0 = - 20 C + 35$ .

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Step 3.7.1

Rewrite the equation as $- 20 C + 35 = 0$ .

$- 20 C + 35 = 0$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.2

Subtract $35$ from both sides of the equation.

$- 20 C = - 35$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3

Divide each term in $- 20 C = - 35$ by $- 20$ and simplify.

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Step 3.7.3.1

Divide each term in $- 20 C = - 35$ by $- 20$ .

$\frac{- 20 C}{- 20} = \frac{- 35}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

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.

$\frac{- 20 C}{- 20} = \frac{- 35}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3.2.1.2

Divide $C$ by $1$ .

$C = \frac{- 35}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{- 35}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{- 35}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3.3

Simplify the right side.

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Step 3.7.3.3.1

Cancel the common factor of $- 35$ and $- 20$ .

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Step 3.7.3.3.1.1

Factor $- 5$ out of $- 35$ .

$C = \frac{- 5 \cdot 7}{- 20}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3.3.1.2

Cancel the common factors.

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Step 3.7.3.3.1.2.1

Factor $- 5$ out of $- 20$ .

$C = \frac{- 5 \cdot 7}{- 5 \cdot 4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3.3.1.2.2

Cancel the common factor.

$C = \frac{- 5 \cdot 7}{- 5 \cdot 4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.7.3.3.1.2.3

Rewrite the expression.

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$A = - C + \frac{7}{2}$

$D = - \frac{7}{20}$

Step 3.8

Replace all occurrences of $C$ with $\frac{7}{4}$ in each equation.

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Step 3.8.1

Replace all occurrences of $C$ in $A = - C + \frac{7}{2}$ with $\frac{7}{4}$ .

$A = - (\frac{7}{4}) + \frac{7}{2}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2

Simplify the right side.

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Step 3.8.2.1

Simplify $- (\frac{7}{4}) + \frac{7}{2}$ .

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Step 3.8.2.1.1

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A = - \frac{7}{4} + \frac{7}{2} \cdot \frac{2}{2}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2.1.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.8.2.1.2.1

Multiply $\frac{7}{2}$ by $\frac{2}{2}$ .

$A = - \frac{7}{4} + \frac{7 \cdot 2}{2 \cdot 2}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2.1.2.2

Multiply $2$ by $2$ .

$A = - \frac{7}{4} + \frac{7 \cdot 2}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

$A = - \frac{7}{4} + \frac{7 \cdot 2}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2.1.3

Combine the numerators over the common denominator.

$A = \frac{- 7 + 7 \cdot 2}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2.1.4

Simplify the numerator.

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Step 3.8.2.1.4.1

Multiply $7$ by $2$ .

$A = \frac{- 7 + 14}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.8.2.1.4.2

Add $- 7$ and $14$ .

$A = \frac{7}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

$A = \frac{7}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

$A = \frac{7}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

$A = \frac{7}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

$A = \frac{7}{4}$

$C = \frac{7}{4}$

$B = \frac{7}{20}$

$D = - \frac{7}{20}$

Step 3.9

List all of the solutions.

$A = \frac{7}{4}, C = \frac{7}{4}, B = \frac{7}{20}, D = - \frac{7}{20}$

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 $D$ .

$\frac{\frac{7}{4}}{{(x - 5)}^{2}} + \frac{\frac{7}{20}}{x - 5} + \frac{\frac{7}{4}}{{(x + 5)}^{2}} + \frac{- \frac{7}{20}}{x + 5}$