Solve for z 8/z-z=4

$\frac{8}{z} - z = 4$

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.

$z, 1, 1$

The LCM of one and any expression is the expression.

$z$

Step 2

Multiply each term in $\frac{8}{z} - z = 4$ by $z$ to eliminate the fractions.

Multiply each term in $\frac{8}{z} - z = 4$ by $z$ .

$\frac{8}{z} z - z \cdot z = 4 z$

Simplify the left side.

Simplify each term.

Cancel the common factor of $z$ .

Cancel the common factor.

$\frac{8}{z} z - z \cdot z = 4 z$

Rewrite the expression.

$8 - z \cdot z = 4 z$

Multiply $z$ by $z$ by adding the exponents.

Move $z$ .

$8 - (z \cdot z) = 4 z$

Multiply $z$ by $z$ .

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

Step 3

Solve the equation.

Subtract $4 z$ from both sides of the equation.

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

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 1$ , $b = - 4$ , and $c = 8$ into the quadratic formula and solve for $z$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot 8)}}{2 \cdot - 1}$

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$z = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot 8}}{2 \cdot - 1}$

Multiply $- 4 \cdot - 1 \cdot 8$ .

Multiply $- 4$ by $- 1$ .

$z = \frac{4 \pm \sqrt{16 + 4 \cdot 8}}{2 \cdot - 1}$

Multiply $4$ by $8$ .

$z = \frac{4 \pm \sqrt{16 + 32}}{2 \cdot - 1}$

Add $16$ and $32$ .

$z = \frac{4 \pm \sqrt{48}}{2 \cdot - 1}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$z = \frac{4 \pm \sqrt{16 (3)}}{2 \cdot - 1}$

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

$z = \frac{4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Pull terms out from under the radical.

$z = \frac{4 \pm 4 \sqrt{3}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$z = \frac{4 \pm 4 \sqrt{3}}{- 2}$

Simplify $\frac{4 \pm 4 \sqrt{3}}{- 2}$ .

$z = \frac{2 \pm 2 \sqrt{3}}{- 1}$

Move the negative one from the denominator of $\frac{2 \pm 2 \sqrt{3}}{- 1}$ .

$z = - 1 \cdot (2 \pm 2 \sqrt{3})$

Rewrite $- 1 \cdot (2 \pm 2 \sqrt{3})$ as $- (2 \pm 2 \sqrt{3})$ .

$z = - (2 \pm 2 \sqrt{3})$

The final answer is the combination of both solutions.

$z = - 2 - 2 \sqrt{3}, - 2 + 2 \sqrt{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$z = - 2 - 2 \sqrt{3}, - 2 + 2 \sqrt{3}$

Decimal Form:

$z = - 5.46410161 \dots, 1.46410161 \dots$

Solve for z 8/z-z=4

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