Algebra Examples

$\frac{x^{2} - 16}{x - 2} \div \frac{x^{2} + 3 x - 4}{x - 8}$

Step 1

Set the denominator in $\frac{x^{2} - 16}{x - 2}$ equal to $0$ to find where the expression is undefined.

$x - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$x = 2$

Step 3

Set the denominator in $\frac{x^{2} + 3 x - 4}{x - 8}$ equal to $0$ to find where the expression is undefined.

$x - 8 = 0$

Step 4

Add $8$ to both sides of the equation.

$x = 8$

Step 5

Set the denominator in $\frac{x^{2} - 16}{x - 2} \div \frac{x^{2} + 3 x - 4}{x - 8}$ equal to $0$ to find where the expression is undefined.

$\frac{x^{2} + 3 x - 4}{x - 8} = 0$

Step 6

Solve for $x$ .

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Step 6.1

Set the numerator equal to zero.

$x^{2} + 3 x - 4 = 0$

Step 6.2

Solve the equation for $x$ .

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Step 6.2.1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 6.2.1.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 6.2.1.2

Write the factored form using these integers.

$(x - 1) (x + 4) = 0$

Step 6.2.2

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

$x - 1 = 0$

$x + 4 = 0$

Step 6.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 6.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 6.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 6.2.4

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 6.2.4.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 6.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 6.2.5

The final solution is all the values that make $(x - 1) (x + 4) = 0$ true.

$x = 1, - 4$

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 $0$ .

$x = - 4, x = 1, x = 2, x = 8$

Step 8