Algebra Examples

$z + \frac{16}{z + 2} = 6$

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.

$1, z + 2, 1$

Step 1.2

Remove parentheses.

$1, z + 2, 1$

Step 1.3

The LCM of one and any expression is the expression.

$z + 2$

Step 2

Multiply each term in $z + \frac{16}{z + 2} = 6$ by $z + 2$ to eliminate the fractions.

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Step 2.1

Multiply each term in $z + \frac{16}{z + 2} = 6$ by $z + 2$ .

$z (z + 2) + \frac{16}{z + 2} (z + 2) = 6 (z + 2)$

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.

$z \cdot z + z \cdot 2 + \frac{16}{z + 2} (z + 2) = 6 (z + 2)$

Step 2.2.1.2

Multiply $z$ by $z$ .

$z^{2} + z \cdot 2 + \frac{16}{z + 2} (z + 2) = 6 (z + 2)$

Step 2.2.1.3

Move $2$ to the left of $z$ .

$z^{2} + 2 \cdot z + \frac{16}{z + 2} (z + 2) = 6 (z + 2)$

Step 2.2.1.4

Cancel the common factor of $z + 2$ .

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Step 2.2.1.4.1

Cancel the common factor.

$z^{2} + 2 z + \frac{16}{z + 2} (z + 2) = 6 (z + 2)$

Step 2.2.1.4.2

Rewrite the expression.

$z^{2} + 2 z + 16 = 6 (z + 2)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Apply the distributive property.

$z^{2} + 2 z + 16 = 6 z + 6 \cdot 2$

Step 2.3.2

Multiply $6$ by $2$ .

$z^{2} + 2 z + 16 = 6 z + 12$

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 $6 z$ from both sides of the equation.

$z^{2} + 2 z + 16 - 6 z = 12$

Step 3.1.2

Subtract $6 z$ from $2 z$ .

$z^{2} - 4 z + 16 = 12$

Step 3.2

Subtract $12$ from both sides of the equation.

$z^{2} - 4 z + 16 - 12 = 0$

Step 3.3

Subtract $12$ from $16$ .

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

Step 3.4

Factor using the perfect square rule.

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Step 3.4.1

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

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

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.

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

Step 3.4.3

Rewrite the polynomial.

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

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 $b = 2$ .

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

Step 3.5

Set the $z - 2$ equal to $0$ .

$z - 2 = 0$

Step 3.6

Add $2$ to both sides of the equation.

$z = 2$