Solve by Factoring z+7/z=-8

$z + \frac{7}{z} = - 8$

Step 1

Add $8$ to both sides of the equation.

$z + \frac{7}{z} + 8 = 0$

Step 2

To write $z$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{z \cdot z}{z} + \frac{7}{z} + 8 = 0$

Step 3

Combine the numerators over the common denominator.

$\frac{z \cdot z + 7}{z} + 8 = 0$

Step 4

Multiply $z$ by $z$ .

$\frac{z^{2} + 7}{z} + 8 = 0$

Step 5

To write $8$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{z^{2} + 7}{z} + \frac{8 z}{z} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{z^{2} + 7 + 8 z}{z} = 0$

Step 7

Factor $z^{2} + 7 + 8 z$ using the AC method.

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 $7$ and whose sum is $8$ .

$1, 7$

Write the factored form using these integers.

$\frac{(z + 1) (z + 7)}{z} = 0$

Step 8

Set the numerator equal to zero.

$(z + 1) (z + 7) = 0$

Step 9

Solve the equation for $z$ .

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

$z + 1 = 0$

$z + 7 = 0$

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

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

$z + 1 = 0$

Subtract $1$ from both sides of the equation.

$z = - 1$

Set $z + 7$ equal to $0$ and solve for $z$ .

Set $z + 7$ equal to $0$ .

$z + 7 = 0$

Subtract $7$ from both sides of the equation.

$z = - 7$

The final solution is all the values that make $(z + 1) (z + 7) = 0$ true.

$z = - 1, - 7$

Solve by Factoring z+7/z=-8

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