Algebra Examples

$\frac{4 - x}{x^{2} - 3 x} - \frac{2 x + 7}{x^{2} - 6 x + 9}$

Step 1

Simplify each term.

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Step 1.1

Factor $x$ out of $x^{2} - 3 x$ .

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Step 1.1.1

Factor $x$ out of $x^{2}$ .

$\frac{4 - x}{x \cdot x - 3 x} - \frac{2 x + 7}{x^{2} - 6 x + 9}$

Step 1.1.2

Factor $x$ out of $- 3 x$ .

$\frac{4 - x}{x \cdot x + x \cdot - 3} - \frac{2 x + 7}{x^{2} - 6 x + 9}$

Step 1.1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

$\frac{4 - x}{x (x - 3)} - \frac{2 x + 7}{x^{2} - 6 x + 9}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $9$ as $3^{2}$ .

$\frac{4 - x}{x (x - 3)} - \frac{2 x + 7}{x^{2} - 6 x + 3^{2}}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.3

Rewrite the polynomial.

$\frac{4 - x}{x (x - 3)} - \frac{2 x + 7}{x^{2} - 2 \cdot x \cdot 3 + 3^{2}}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

$\frac{4 - x}{x (x - 3)} - \frac{2 x + 7}{{(x - 3)}^{2}}$

Step 2

To write $\frac{4 - x}{x (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

$\frac{4 - x}{x (x - 3)} \cdot \frac{x - 3}{x - 3} - \frac{2 x + 7}{{(x - 3)}^{2}}$

Step 3

To write $- \frac{2 x + 7}{{(x - 3)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 - x}{x (x - 3)} \cdot \frac{x - 3}{x - 3} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4

Write each expression with a common denominator of $x {(x - 3)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{4 - x}{x (x - 3)}$ by $\frac{x - 3}{x - 3}$ .

$\frac{(4 - x) (x - 3)}{x (x - 3) (x - 3)} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4.2

Raise $x - 3$ to the power of $1$ .

$\frac{(4 - x) (x - 3)}{x ({(x - 3)}^{1} (x - 3))} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4.3

Raise $x - 3$ to the power of $1$ .

$\frac{(4 - x) (x - 3)}{x ({(x - 3)}^{1} {(x - 3)}^{1})} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4.4

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

$\frac{(4 - x) (x - 3)}{x {(x - 3)}^{1 + 1}} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4.5

Add $1$ and $1$ .

$\frac{(4 - x) (x - 3)}{x {(x - 3)}^{2}} - \frac{2 x + 7}{{(x - 3)}^{2}} \cdot \frac{x}{x}$

Step 4.6

Multiply $\frac{2 x + 7}{{(x - 3)}^{2}}$ by $\frac{x}{x}$ .

$\frac{(4 - x) (x - 3)}{x {(x - 3)}^{2}} - \frac{(2 x + 7) x}{{(x - 3)}^{2} x}$

Step 4.7

Reorder the factors of ${(x - 3)}^{2} x$ .

$\frac{(4 - x) (x - 3)}{x {(x - 3)}^{2}} - \frac{(2 x + 7) x}{x {(x - 3)}^{2}}$

Step 5

Combine the numerators over the common denominator.

$\frac{(4 - x) (x - 3) - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6

Simplify the numerator.

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Step 6.1

Expand $(4 - x) (x - 3)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$\frac{4 (x - 3) - x (x - 3) - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.1.2

Apply the distributive property.

$\frac{4 x + 4 \cdot - 3 - x (x - 3) - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.1.3

Apply the distributive property.

$\frac{4 x + 4 \cdot - 3 - x \cdot x - x \cdot - 3 - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $4$ by $- 3$ .

$\frac{4 x - 12 - x \cdot x - x \cdot - 3 - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.2.1.2

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

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Step 6.2.1.2.1

Move $x$ .

$\frac{4 x - 12 - (x \cdot x) - x \cdot - 3 - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.2.1.2.2

Multiply $x$ by $x$ .

$\frac{4 x - 12 - x^{2} - x \cdot - 3 - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.2.1.3

Multiply $- 3$ by $- 1$ .

$\frac{4 x - 12 - x^{2} + 3 x - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.2.2

Add $4 x$ and $3 x$ .

$\frac{7 x - 12 - x^{2} - (2 x + 7) x}{x {(x - 3)}^{2}}$

Step 6.3

Apply the distributive property.

$\frac{7 x - 12 - x^{2} + (- (2 x) - 1 \cdot 7) x}{x {(x - 3)}^{2}}$

Step 6.4

Multiply $2$ by $- 1$ .

$\frac{7 x - 12 - x^{2} + (- 2 x - 1 \cdot 7) x}{x {(x - 3)}^{2}}$

Step 6.5

Multiply $- 1$ by $7$ .

$\frac{7 x - 12 - x^{2} + (- 2 x - 7) x}{x {(x - 3)}^{2}}$

Step 6.6

Apply the distributive property.

$\frac{7 x - 12 - x^{2} - 2 x \cdot x - 7 x}{x {(x - 3)}^{2}}$

Step 6.7

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

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Step 6.7.1

Move $x$ .

$\frac{7 x - 12 - x^{2} - 2 (x \cdot x) - 7 x}{x {(x - 3)}^{2}}$

Step 6.7.2

Multiply $x$ by $x$ .

$\frac{7 x - 12 - x^{2} - 2 x^{2} - 7 x}{x {(x - 3)}^{2}}$

Step 6.8

Subtract $7 x$ from $7 x$ .

$\frac{- 12 - x^{2} - 2 x^{2} + 0}{x {(x - 3)}^{2}}$

Step 6.9

Add $- 12 - x^{2} - 2 x^{2}$ and $0$ .

$\frac{- 12 - x^{2} - 2 x^{2}}{x {(x - 3)}^{2}}$

Step 6.10

Subtract $2 x^{2}$ from $- x^{2}$ .

$\frac{- 12 - 3 x^{2}}{x {(x - 3)}^{2}}$

Step 6.11

Factor $3$ out of $- 12 - 3 x^{2}$ .

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Step 6.11.1

Factor $3$ out of $- 12$ .

$\frac{3 (- 4) - 3 x^{2}}{x {(x - 3)}^{2}}$

Step 6.11.2

Factor $3$ out of $- 3 x^{2}$ .

$\frac{3 (- 4) + 3 (- x^{2})}{x {(x - 3)}^{2}}$

Step 6.11.3

Factor $3$ out of $3 (- 4) + 3 (- x^{2})$ .

$\frac{3 (- 4 - x^{2})}{x {(x - 3)}^{2}}$

Step 7

Simplify with factoring out.

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Step 7.1

Rewrite $- 4$ as $- 1 (4)$ .

$\frac{3 (- 1 (4) - x^{2})}{x {(x - 3)}^{2}}$

Step 7.2

Factor $- 1$ out of $- x^{2}$ .

$\frac{3 (- 1 (4) - (x^{2}))}{x {(x - 3)}^{2}}$

Step 7.3

Factor $- 1$ out of $- 1 (4) - (x^{2})$ .

$\frac{3 (- 1 (4 + x^{2}))}{x {(x - 3)}^{2}}$

Step 7.4

Move the negative in front of the fraction.

$- \frac{3 (4 + x^{2})}{x {(x - 3)}^{2}}$