Algebra Examples

$\frac{x^{2} (x^{2} - 5 x - 14)}{4 x (x^{2} + 6 x + 8)}$

Step 1

Set the denominator in $\frac{x^{2} (x^{2} - 5 x - 14)}{4 x (x^{2} + 6 x + 8)}$ equal to $0$ to find where the expression is undefined.

$4 x (x^{2} + 6 x + 8) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x = 0$

$x^{2} + 6 x + 8 = 0$

Step 2.2

Set $x$ equal to $0$ .

$x = 0$

Step 2.3

Set $x^{2} + 6 x + 8$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 6 x + 8$ equal to $0$ .

$x^{2} + 6 x + 8 = 0$

Step 2.3.2

Solve $x^{2} + 6 x + 8 = 0$ for $x$ .

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Step 2.3.2.1

Factor $x^{2} + 6 x + 8$ using the AC method.

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Step 2.3.2.1.1

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 $8$ and whose sum is $6$ .

$2, 4$

Step 2.3.2.1.2

Write the factored form using these integers.

$(x + 2) (x + 4) = 0$

Step 2.3.2.2

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

$x + 2 = 0$

$x + 4 = 0$

Step 2.3.2.3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.3.2.3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.3.2.3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.3.2.4

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

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Step 2.3.2.4.1

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

$x + 4 = 0$

Step 2.3.2.4.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.3.2.5

The final solution is all the values that make $(x + 2) (x + 4) = 0$ true.

$x = - 2, - 4$

Step 2.4

The final solution is all the values that make $4 x (x^{2} + 6 x + 8) = 0$ true.

$x = 0, - 2, - 4$

Step 3

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 $0$ .

$x = - 4, x = - 2, x = 0$

Step 4