Simplify (x^2+4x+4)/(x^2+9x+14)

$\frac{x^{2} + 4 x + 4}{x^{2} + 9 x + 14}$

Step 1

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

$\frac{x^{2} + 4 x + 2^{2}}{x^{2} + 9 x + 14}$

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

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

$\frac{x^{2} + 2 \cdot x \cdot 2 + 2^{2}}{x^{2} + 9 x + 14}$

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

$\frac{{(x + 2)}^{2}}{x^{2} + 9 x + 14}$

Step 2

Factor $x^{2} + 9 x + 14$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $14$ and whose sum is $9$ .

$2, 7$

Write the factored form using these integers.

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

Step 3

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

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

$\frac{(x + 2) (x + 2)}{(x + 2) (x + 7)}$

Cancel the common factors.

Cancel the common factor.

$\frac{(x + 2) (x + 2)}{(x + 2) (x + 7)}$

Rewrite the expression.

$\frac{x + 2}{x + 7}$

Simplify (x^2+4x+4)/(x^2+9x+14)

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