Find All Complex Solutions z+4/(z+1)=3

$z + \frac{4}{z + 1} = 3$

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.

$1, z + 1, 1$

Remove parentheses.

$1, z + 1, 1$

The LCM of one and any expression is the expression.

$z + 1$

Step 2

Multiply each term in $z + \frac{4}{z + 1} = 3$ by $z + 1$ to eliminate the fractions.

Multiply each term in $z + \frac{4}{z + 1} = 3$ by $z + 1$ .

$z (z + 1) + \frac{4}{z + 1} (z + 1) = 3 (z + 1)$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$z \cdot z + z \cdot 1 + \frac{4}{z + 1} (z + 1) = 3 (z + 1)$

Multiply $z$ by $z$ .

$z^{2} + z \cdot 1 + \frac{4}{z + 1} (z + 1) = 3 (z + 1)$

Multiply $z$ by $1$ .

$z^{2} + z + \frac{4}{z + 1} (z + 1) = 3 (z + 1)$

Cancel the common factor of $z + 1$ .

Cancel the common factor.

$z^{2} + z + \frac{4}{z + 1} (z + 1) = 3 (z + 1)$

Rewrite the expression.

$z^{2} + z + 4 = 3 (z + 1)$

Simplify the right side.

Apply the distributive property.

$z^{2} + z + 4 = 3 z + 3 \cdot 1$

Multiply $3$ by $1$ .

$z^{2} + z + 4 = 3 z + 3$

Step 3

Solve the equation.

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

Subtract $3 z$ from both sides of the equation.

$z^{2} + z + 4 - 3 z = 3$

Subtract $3 z$ from $z$ .

$z^{2} - 2 z + 4 = 3$

Subtract $3$ from both sides of the equation.

$z^{2} - 2 z + 4 - 3 = 0$

Subtract $3$ from $4$ .

$z^{2} - 2 z + 1 = 0$

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

$z^{2} - 2 z + 1^{2} = 0$

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

$2 z = 2 \cdot z \cdot 1$

Rewrite the polynomial.

$z^{2} - 2 \cdot z \cdot 1 + 1^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = z$ and $b = 1$ .

${(z - 1)}^{2} = 0$

Set the $z - 1$ equal to $0$ .

$z - 1 = 0$

Add $1$ to both sides of the equation.

$z = 1$

Find All Complex Solutions z+4/(z+1)=3

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