Basic Math Examples

$\frac{t^{2} + 3 t}{t^{2} + 13 t + 42} - \frac{t + 35}{t^{2} + 13 t + 42}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{t^{2} + 3 t - (t + 35)}{t^{2} + 13 t + 42}$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{t^{2} + 3 t - t - 1 \cdot 35}{t^{2} + 13 t + 42}$

Step 1.2.2

Multiply $- 1$ by $35$ .

$\frac{t^{2} + 3 t - t - 35}{t^{2} + 13 t + 42}$

Step 1.3

Subtract $t$ from $3 t$ .

$\frac{t^{2} + 2 t - 35}{t^{2} + 13 t + 42}$

Step 2

Factor $t^{2} + 2 t - 35$ using the AC method.

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Step 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 $- 35$ and whose sum is $2$ .

$- 5, 7$

Step 2.2

Write the factored form using these integers.

$\frac{(t - 5) (t + 7)}{t^{2} + 13 t + 42}$

Step 3

Factor $t^{2} + 13 t + 42$ using the AC method.

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Step 3.1

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 $42$ and whose sum is $13$ .

$6, 7$

Step 3.2

Write the factored form using these integers.

$\frac{(t - 5) (t + 7)}{(t + 6) (t + 7)}$

Step 4

Cancel the common factor of $t + 7$ .

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Step 4.1

Cancel the common factor.

$\frac{(t - 5) (t + 7)}{(t + 6) (t + 7)}$

Step 4.2

Rewrite the expression.

$\frac{t - 5}{t + 6}$