Algebra Examples

$\frac{a + 2}{a - 5} \div \frac{a + 1}{a^{2} - 8 a + 15}$

Step 1

Set the denominator in $\frac{a + 2}{a - 5}$ equal to $0$ to find where the expression is undefined.

$a - 5 = 0$

Step 2

Add $5$ to both sides of the equation.

$a = 5$

Step 3

Set the denominator in $\frac{a + 1}{a^{2} - 8 a + 15}$ equal to $0$ to find where the expression is undefined.

$a^{2} - 8 a + 15 = 0$

Step 4

Solve for $a$ .

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Step 4.1

Factor $a^{2} - 8 a + 15$ using the AC method.

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Step 4.1.1

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 $15$ and whose sum is $- 8$ .

$- 5, - 3$

Step 4.1.2

Write the factored form using these integers.

$(a - 5) (a - 3) = 0$

Step 4.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a - 5 = 0$

$a - 3 = 0$

Step 4.3

Set $a - 5$ equal to $0$ and solve for $a$ .

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Step 4.3.1

Set $a - 5$ equal to $0$ .

$a - 5 = 0$

Step 4.3.2

Add $5$ to both sides of the equation.

$a = 5$

Step 4.4

Set $a - 3$ equal to $0$ and solve for $a$ .

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Step 4.4.1

Set $a - 3$ equal to $0$ .

$a - 3 = 0$

Step 4.4.2

Add $3$ to both sides of the equation.

$a = 3$

Step 4.5

The final solution is all the values that make $(a - 5) (a - 3) = 0$ true.

$a = 5, 3$

Step 5

Set the denominator in $\frac{a + 2}{a - 5} \div \frac{a + 1}{a^{2} - 8 a + 15}$ equal to $0$ to find where the expression is undefined.

$\frac{a + 1}{a^{2} - 8 a + 15} = 0$

Step 6

Solve for $a$ .

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Step 6.1

Set the numerator equal to zero.

$a + 1 = 0$

Step 6.2

Subtract $1$ from both sides of the equation.

$a = - 1$

Step 7

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$a = - 1, a = 3, a = 5$

Step 8