Precalculus Examples

$z + \frac{8}{z} = - 9$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, z, 1$

Step 1.2

The LCM of one and any expression is the expression.

$z$

Step 2

Multiply each term in $z + \frac{8}{z} = - 9$ by $z$ to eliminate the fractions.

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Step 2.1

Multiply each term in $z + \frac{8}{z} = - 9$ by $z$ .

$z \cdot z + \frac{8}{z} z = - 9 z$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $z$ by $z$ .

$z^{2} + \frac{8}{z} z = - 9 z$

Step 2.2.1.2

Cancel the common factor of $z$ .

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Step 2.2.1.2.1

Cancel the common factor.

$z^{2} + \frac{8}{z} z = - 9 z$

Step 2.2.1.2.2

Rewrite the expression.

$z^{2} + 8 = - 9 z$

Step 3

Solve the equation.

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Step 3.1

Add $9 z$ to both sides of the equation.

$z^{2} + 8 + 9 z = 0$

Step 3.2

Factor $z^{2} + 8 + 9 z$ using the AC method.

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Step 3.2.1

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 $8$ and whose sum is $9$ .

$1, 8$

Step 3.2.2

Write the factored form using these integers.

$(z + 1) (z + 8) = 0$

Step 3.3

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

$z + 1 = 0$

$z + 8 = 0$

Step 3.4

Set $z + 1$ equal to $0$ and solve for $z$ .

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Step 3.4.1

Set $z + 1$ equal to $0$ .

$z + 1 = 0$

Step 3.4.2

Subtract $1$ from both sides of the equation.

$z = - 1$

Step 3.5

Set $z + 8$ equal to $0$ and solve for $z$ .

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Step 3.5.1

Set $z + 8$ equal to $0$ .

$z + 8 = 0$

Step 3.5.2

Subtract $8$ from both sides of the equation.

$z = - 8$

Step 3.6

The final solution is all the values that make $(z + 1) (z + 8) = 0$ true.

$z = - 1, - 8$