Basic Math Examples

$\frac{8}{z - 2} = \frac{z + 2}{4}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$8 \cdot 4 = (z - 2) (z + 2)$

Step 2

Solve the equation for $z$ .

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Step 2.1

Rewrite the equation as $(z - 2) (z + 2) = 8 \cdot 4$ .

$(z - 2) (z + 2) = 8 \cdot 4$

Step 2.2

Simplify $(z - 2) (z + 2)$ .

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Step 2.2.1

Expand $(z - 2) (z + 2)$ using the FOIL Method.

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Step 2.2.1.1

Apply the distributive property.

$z (z + 2) - 2 (z + 2) = 8 \cdot 4$

Step 2.2.1.2

Apply the distributive property.

$z \cdot z + z \cdot 2 - 2 (z + 2) = 8 \cdot 4$

Step 2.2.1.3

Apply the distributive property.

$z \cdot z + z \cdot 2 - 2 z - 2 \cdot 2 = 8 \cdot 4$

Step 2.2.2

Simplify terms.

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Step 2.2.2.1

Combine the opposite terms in $z \cdot z + z \cdot 2 - 2 z - 2 \cdot 2$ .

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Step 2.2.2.1.1

Reorder the factors in the terms $z \cdot 2$ and $- 2 z$ .

$z \cdot z + 2 z - 2 z - 2 \cdot 2 = 8 \cdot 4$

Step 2.2.2.1.2

Subtract $2 z$ from $2 z$ .

$z \cdot z + 0 - 2 \cdot 2 = 8 \cdot 4$

Step 2.2.2.1.3

Add $z \cdot z$ and $0$ .

$z \cdot z - 2 \cdot 2 = 8 \cdot 4$

Step 2.2.2.2

Simplify each term.

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Step 2.2.2.2.1

Multiply $z$ by $z$ .

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

Step 2.2.2.2.2

Multiply $- 2$ by $2$ .

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

Step 2.3

Multiply $8$ by $4$ .

$z^{2} - 4 = 32$

Step 2.4

Move all terms not containing $z$ to the right side of the equation.

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Step 2.4.1

Add $4$ to both sides of the equation.

$z^{2} = 32 + 4$

Step 2.4.2

Add $32$ and $4$ .

$z^{2} = 36$

Step 2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{36}$

Step 2.6

Simplify $\pm \sqrt{36}$ .

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Step 2.6.1

Rewrite $36$ as $6^{2}$ .

$z = \pm \sqrt{6^{2}}$

Step 2.6.2

Pull terms out from under the radical, assuming positive real numbers.

$z = \pm 6$

Step 2.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.7.1

First, use the positive value of the $\pm$ to find the first solution.

$z = 6$

Step 2.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$z = - 6$

Step 2.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$z = 6, - 6$