Solve for z z+36/(z+3)=9

$z + \frac{36}{z + 3} = 9$

Step 1

Find the LCD of the terms in the equation.

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

$1, z + 3, 1$

Remove parentheses.

$1, z + 3, 1$

The LCM of one and any expression is the expression.

$z + 3$

Step 2

Multiply each term in $z + \frac{36}{z + 3} = 9$ by $z + 3$ to eliminate the fractions.

Multiply each term in $z + \frac{36}{z + 3} = 9$ by $z + 3$ .

$z (z + 3) + \frac{36}{z + 3} (z + 3) = 9 (z + 3)$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$z \cdot z + z \cdot 3 + \frac{36}{z + 3} (z + 3) = 9 (z + 3)$

Multiply $z$ by $z$ .

$z^{2} + z \cdot 3 + \frac{36}{z + 3} (z + 3) = 9 (z + 3)$

Move $3$ to the left of $z$ .

$z^{2} + 3 \cdot z + \frac{36}{z + 3} (z + 3) = 9 (z + 3)$

Cancel the common factor of $z + 3$ .

Cancel the common factor.

$z^{2} + 3 z + \frac{36}{z + 3} (z + 3) = 9 (z + 3)$

Rewrite the expression.

$z^{2} + 3 z + 36 = 9 (z + 3)$

Simplify the right side.

Apply the distributive property.

$z^{2} + 3 z + 36 = 9 z + 9 \cdot 3$

Multiply $9$ by $3$ .

$z^{2} + 3 z + 36 = 9 z + 27$

Step 3

Solve the equation.

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

Subtract $9 z$ from both sides of the equation.

$z^{2} + 3 z + 36 - 9 z = 27$

Subtract $9 z$ from $3 z$ .

$z^{2} - 6 z + 36 = 27$

Subtract $27$ from both sides of the equation.

$z^{2} - 6 z + 36 - 27 = 0$

Subtract $27$ from $36$ .

$z^{2} - 6 z + 9 = 0$

Factor using the perfect square rule.

Rewrite $9$ as $3^{2}$ .

$z^{2} - 6 z + 3^{2} = 0$

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

$6 z = 2 \cdot z \cdot 3$

Rewrite the polynomial.

$z^{2} - 2 \cdot z \cdot 3 + 3^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 3$ .

${(z - 3)}^{2} = 0$

Set the $z - 3$ equal to $0$ .

$z - 3 = 0$

Add $3$ to both sides of the equation.

$z = 3$

Solve for z z+36/(z+3)=9

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