Simplify ((x-5)^2-12)/((x-5)^2+4)

$\frac{{(x - 5)}^{2} - 12}{{(x - 5)}^{2} + 4}$

Step 1

Simplify the numerator.

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

$\frac{(x - 5) (x - 5) - 12}{{(x - 5)}^{2} + 4}$

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

Apply the distributive property.

$\frac{x (x - 5) - 5 (x - 5) - 12}{{(x - 5)}^{2} + 4}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 - 5 (x - 5) - 12}{{(x - 5)}^{2} + 4}$

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

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

$\frac{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 12}{{(x - 5)}^{2} + 4}$

Multiply $- 5$ by $- 5$ .

$\frac{x^{2} - 5 x - 5 x + 25 - 12}{{(x - 5)}^{2} + 4}$

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

$\frac{x^{2} - 10 x + 25 - 12}{{(x - 5)}^{2} + 4}$

Subtract $12$ from $25$ .

$\frac{x^{2} - 10 x + 13}{{(x - 5)}^{2} + 4}$

Step 2

Simplify the denominator.

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

$\frac{x^{2} - 10 x + 13}{(x - 5) (x - 5) + 4}$

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

Apply the distributive property.

$\frac{x^{2} - 10 x + 13}{x (x - 5) - 5 (x - 5) + 4}$

Apply the distributive property.

$\frac{x^{2} - 10 x + 13}{x \cdot x + x \cdot - 5 - 5 (x - 5) + 4}$

Apply the distributive property.

$\frac{x^{2} - 10 x + 13}{x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 + 4}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$\frac{x^{2} - 10 x + 13}{x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 + 4}$

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

$\frac{x^{2} - 10 x + 13}{x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 + 4}$

Multiply $- 5$ by $- 5$ .

$\frac{x^{2} - 10 x + 13}{x^{2} - 5 x - 5 x + 25 + 4}$

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

$\frac{x^{2} - 10 x + 13}{x^{2} - 10 x + 25 + 4}$

Add $25$ and $4$ .

$\frac{x^{2} - 10 x + 13}{x^{2} - 10 x + 29}$

Simplify ((x-5)^2-12)/((x-5)^2+4)

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