Simplify (n^2-10n+24)/(n^2-13n+42)-9/(n-7)

$\frac{n^{2} - 10 n + 24}{n^{2} - 13 n + 42} - \frac{9}{n - 7}$

Step 1

Simplify each term.

Factor $n^{2} - 10 n + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Write the factored form using these integers.

$\frac{(n - 6) (n - 4)}{n^{2} - 13 n + 42} - \frac{9}{n - 7}$

Factor $n^{2} - 13 n + 42$ 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 $42$ and whose sum is $- 13$ .

$- 7, - 6$

Write the factored form using these integers.

$\frac{(n - 6) (n - 4)}{(n - 7) (n - 6)} - \frac{9}{n - 7}$

Cancel the common factor of $n - 6$ .

Cancel the common factor.

$\frac{(n - 6) (n - 4)}{(n - 7) (n - 6)} - \frac{9}{n - 7}$

Rewrite the expression.

$\frac{n - 4}{n - 7} - \frac{9}{n - 7}$

Step 2

Combine fractions.

Combine the numerators over the common denominator.

$\frac{n - 4 - 9}{n - 7}$

Subtract $9$ from $- 4$ .

$\frac{n - 13}{n - 7}$

Simplify (n^2-10n+24)/(n^2-13n+42)-9/(n-7)

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