Pre-Algebra Examples

$\frac{7}{x^{2} - 10 x + 16} + \frac{2 x}{x^{2} - 13 x + 40}$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Factor $x^{2} - 10 x + 16$ using the AC method.

Tap for more steps...

Step 1.1.1

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 $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 1.1.2

Write the factored form using these integers.

$\frac{7}{(x - 8) (x - 2)} + \frac{2 x}{x^{2} - 13 x + 40}$

Step 1.2

Factor $x^{2} - 13 x + 40$ using the AC method.

Tap for more steps...

Step 1.2.1

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 $40$ and whose sum is $- 13$ .

$- 8, - 5$

Step 1.2.2

Write the factored form using these integers.

$\frac{7}{(x - 8) (x - 2)} + \frac{2 x}{(x - 8) (x - 5)}$

Step 2

To write $\frac{7}{(x - 8) (x - 2)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{7}{(x - 8) (x - 2)} \cdot \frac{x - 5}{x - 5} + \frac{2 x}{(x - 8) (x - 5)}$

Step 3

To write $\frac{2 x}{(x - 8) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

$\frac{7}{(x - 8) (x - 2)} \cdot \frac{x - 5}{x - 5} + \frac{2 x}{(x - 8) (x - 5)} \cdot \frac{x - 2}{x - 2}$

Step 4

Write each expression with a common denominator of $(x - 8) (x - 2) (x - 5)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{7}{(x - 8) (x - 2)}$ by $\frac{x - 5}{x - 5}$ .

$\frac{7 (x - 5)}{(x - 8) (x - 2) (x - 5)} + \frac{2 x}{(x - 8) (x - 5)} \cdot \frac{x - 2}{x - 2}$

Step 4.2

Multiply $\frac{2 x}{(x - 8) (x - 5)}$ by $\frac{x - 2}{x - 2}$ .

$\frac{7 (x - 5)}{(x - 8) (x - 2) (x - 5)} + \frac{2 x (x - 2)}{(x - 8) (x - 5) (x - 2)}$

Step 4.3

Reorder the factors of $(x - 8) (x - 2) (x - 5)$ .

$\frac{7 (x - 5)}{(x - 2) (x - 5) (x - 8)} + \frac{2 x (x - 2)}{(x - 8) (x - 5) (x - 2)}$

Step 4.4

Reorder the factors of $(x - 8) (x - 5) (x - 2)$ .

$\frac{7 (x - 5)}{(x - 2) (x - 5) (x - 8)} + \frac{2 x (x - 2)}{(x - 2) (x - 5) (x - 8)}$

Step 5

Combine the numerators over the common denominator.

$\frac{7 (x - 5) + 2 x (x - 2)}{(x - 2) (x - 5) (x - 8)}$

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Apply the distributive property.

$\frac{7 x + 7 \cdot - 5 + 2 x (x - 2)}{(x - 2) (x - 5) (x - 8)}$

Step 6.2

Multiply $7$ by $- 5$ .

$\frac{7 x - 35 + 2 x (x - 2)}{(x - 2) (x - 5) (x - 8)}$

Step 6.3

Apply the distributive property.

$\frac{7 x - 35 + 2 x \cdot x + 2 x \cdot - 2}{(x - 2) (x - 5) (x - 8)}$

Step 6.4

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

Tap for more steps...

Step 6.4.1

Move $x$ .

$\frac{7 x - 35 + 2 (x \cdot x) + 2 x \cdot - 2}{(x - 2) (x - 5) (x - 8)}$

Step 6.4.2

Multiply $x$ by $x$ .

$\frac{7 x - 35 + 2 x^{2} + 2 x \cdot - 2}{(x - 2) (x - 5) (x - 8)}$

Step 6.5

Multiply $- 2$ by $2$ .

$\frac{7 x - 35 + 2 x^{2} - 4 x}{(x - 2) (x - 5) (x - 8)}$

Step 6.6

Subtract $4 x$ from $7 x$ .

$\frac{3 x - 35 + 2 x^{2}}{(x - 2) (x - 5) (x - 8)}$

Step 6.7

Reorder terms.

$\frac{2 x^{2} + 3 x - 35}{(x - 2) (x - 5) (x - 8)}$

Step 6.8

Factor by grouping.

Tap for more steps...

Step 6.8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 35 = - 70$ and whose sum is $b = 3$ .

Tap for more steps...

Step 6.8.1.1

Factor $3$ out of $3 x$ .

$\frac{2 x^{2} + 3 (x) - 35}{(x - 2) (x - 5) (x - 8)}$

Step 6.8.1.2

Rewrite $3$ as $- 7$ plus $10$

$\frac{2 x^{2} + (- 7 + 10) x - 35}{(x - 2) (x - 5) (x - 8)}$

Step 6.8.1.3

Apply the distributive property.

$\frac{2 x^{2} - 7 x + 10 x - 35}{(x - 2) (x - 5) (x - 8)}$

Step 6.8.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 6.8.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 7 x) + 10 x - 35}{(x - 2) (x - 5) (x - 8)}$

Step 6.8.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x - 7) + 5 (2 x - 7)}{(x - 2) (x - 5) (x - 8)}$

Step 6.8.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 7$ .

$\frac{(2 x - 7) (x + 5)}{(x - 2) (x - 5) (x - 8)}$