Algebra Examples

$z + \frac{64}{z + 4} = 12$

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.

$1, z + 4, 1$

Step 1.2

Remove parentheses.

$1, z + 4, 1$

Step 1.3

The LCM of one and any expression is the expression.

$z + 4$

Step 2

Multiply each term in $z + \frac{64}{z + 4} = 12$ by $z + 4$ to eliminate the fractions.

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Step 2.1

Multiply each term in $z + \frac{64}{z + 4} = 12$ by $z + 4$ .

$z (z + 4) + \frac{64}{z + 4} (z + 4) = 12 (z + 4)$

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

Apply the distributive property.

$z \cdot z + z \cdot 4 + \frac{64}{z + 4} (z + 4) = 12 (z + 4)$

Step 2.2.1.2

Multiply $z$ by $z$ .

$z^{2} + z \cdot 4 + \frac{64}{z + 4} (z + 4) = 12 (z + 4)$

Step 2.2.1.3

Move $4$ to the left of $z$ .

$z^{2} + 4 \cdot z + \frac{64}{z + 4} (z + 4) = 12 (z + 4)$

Step 2.2.1.4

Cancel the common factor of $z + 4$ .

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Step 2.2.1.4.1

Cancel the common factor.

$z^{2} + 4 z + \frac{64}{z + 4} (z + 4) = 12 (z + 4)$

Step 2.2.1.4.2

Rewrite the expression.

$z^{2} + 4 z + 64 = 12 (z + 4)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$z^{2} + 4 z + 64 = 12 z + 12 \cdot 4$

Step 2.3.2

Multiply $12$ by $4$ .

$z^{2} + 4 z + 64 = 12 z + 48$

Step 3

Solve the equation.

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Step 3.1

Move all terms containing $z$ to the left side of the equation.

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Step 3.1.1

Subtract $12 z$ from both sides of the equation.

$z^{2} + 4 z + 64 - 12 z = 48$

Step 3.1.2

Subtract $12 z$ from $4 z$ .

$z^{2} - 8 z + 64 = 48$

Step 3.2

Subtract $48$ from both sides of the equation.

$z^{2} - 8 z + 64 - 48 = 0$

Step 3.3

Subtract $48$ from $64$ .

$z^{2} - 8 z + 16 = 0$

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

Rewrite $16$ as $4^{2}$ .

$z^{2} - 8 z + 4^{2} = 0$

Step 3.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 z = 2 \cdot z \cdot 4$

Step 3.4.3

Rewrite the polynomial.

$z^{2} - 2 \cdot z \cdot 4 + 4^{2} = 0$

Step 3.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 4$ .

${(z - 4)}^{2} = 0$

Step 3.5

Set the $z - 4$ equal to $0$ .

$z - 4 = 0$

Step 3.6

Add $4$ to both sides of the equation.

$z = 4$