Pre-Algebra Examples

$\frac{2}{z - 5} = \frac{2}{0.5 z^{2} + 7}$

Step 1

Simplify both sides.

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Step 1.1

Factor $0.5$ out of $0.5 z^{2} + 7$ .

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Step 1.1.1

Factor $0.5$ out of $0.5 z^{2}$ .

$\frac{2}{z - 5} = \frac{2}{0.5 (z^{2}) + 7}$

Step 1.1.2

Factor $0.5$ out of $7$ .

$\frac{2}{z - 5} = \frac{2}{0.5 z^{2} + 0.5 \cdot 14}$

Step 1.1.3

Factor $0.5$ out of $0.5 z^{2} + 0.5 \cdot 14$ .

$\frac{2}{z - 5} = \frac{2}{0.5 (z^{2} + 14)}$

Step 1.2

Factor $2$ out of $2$ .

$\frac{2}{z - 5} = \frac{2 (1)}{0.5 (z^{2} + 14)}$

Step 1.3

Separate fractions.

$\frac{2}{z - 5} = \frac{2}{0.5} \cdot \frac{1}{z^{2} + 14}$

Step 1.4

Divide $2$ by $0.5$ .

$\frac{2}{z - 5} = 4 \frac{1}{z^{2} + 14}$

Step 1.5

Combine $4$ and $\frac{1}{z^{2} + 14}$ .

$\frac{2}{z - 5} = \frac{4}{z^{2} + 14}$

Step 2

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.

$2 (z^{2} + 14) = (z - 5) \cdot 4$

Step 3

Solve the equation for $z$ .

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Step 3.1

Simplify $2 (z^{2} + 14)$ .

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Step 3.1.1

Rewrite.

$0 + 0 + 2 (z^{2} + 14) = (z - 5) \cdot 4$

Step 3.1.2

Simplify by adding zeros.

$2 (z^{2} + 14) = (z - 5) \cdot 4$

Step 3.1.3

Apply the distributive property.

$2 z^{2} + 2 \cdot 14 = (z - 5) \cdot 4$

Step 3.1.4

Multiply $2$ by $14$ .

$2 z^{2} + 28 = (z - 5) \cdot 4$

Step 3.2

Simplify $(z - 5) \cdot 4$ .

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Step 3.2.1

Apply the distributive property.

$2 z^{2} + 28 = z \cdot 4 - 5 \cdot 4$

Step 3.2.2

Simplify the expression.

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Step 3.2.2.1

Move $4$ to the left of $z$ .

$2 z^{2} + 28 = 4 \cdot z - 5 \cdot 4$

Step 3.2.2.2

Multiply $- 5$ by $4$ .

$2 z^{2} + 28 = 4 z - 20$

Step 3.3

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

$2 z^{2} + 28 - 4 z = - 20$

Step 3.4

Add $20$ to both sides of the equation.

$2 z^{2} + 28 - 4 z + 20 = 0$

Step 3.5

Add $28$ and $20$ .

$2 z^{2} - 4 z + 48 = 0$

Step 3.6

Factor $2$ out of $2 z^{2} - 4 z + 48$ .

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Step 3.6.1

Factor $2$ out of $2 z^{2}$ .

$2 (z^{2}) - 4 z + 48 = 0$

Step 3.6.2

Factor $2$ out of $- 4 z$ .

$2 (z^{2}) + 2 (- 2 z) + 48 = 0$

Step 3.6.3

Factor $2$ out of $48$ .

$2 z^{2} + 2 (- 2 z) + 2 \cdot 24 = 0$

Step 3.6.4

Factor $2$ out of $2 z^{2} + 2 (- 2 z)$ .

$2 (z^{2} - 2 z) + 2 \cdot 24 = 0$

Step 3.6.5

Factor $2$ out of $2 (z^{2} - 2 z) + 2 \cdot 24$ .

$2 (z^{2} - 2 z + 24) = 0$

Step 3.7

Divide each term in $2 (z^{2} - 2 z + 24) = 0$ by $2$ and simplify.

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Step 3.7.1

Divide each term in $2 (z^{2} - 2 z + 24) = 0$ by $2$ .

$\frac{2 (z^{2} - 2 z + 24)}{2} = \frac{0}{2}$

Step 3.7.2

Simplify the left side.

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Step 3.7.2.1

Cancel the common factor of $2$ .

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Step 3.7.2.1.1

Cancel the common factor.

$\frac{2 (z^{2} - 2 z + 24)}{2} = \frac{0}{2}$

Step 3.7.2.1.2

Divide $z^{2} - 2 z + 24$ by $1$ .

$z^{2} - 2 z + 24 = \frac{0}{2}$

Step 3.7.3

Simplify the right side.

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Step 3.7.3.1

Divide $0$ by $2$ .

$z^{2} - 2 z + 24 = 0$

Step 3.8

Use the quadratic formula to find the solutions.

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

Step 3.9

Substitute the values $a = 1$ , $b = - 2$ , and $c = 24$ into the quadratic formula and solve for $z$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (1 \cdot 24)}}{2 \cdot 1}$

Step 3.10

Simplify.

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Step 3.10.1

Simplify the numerator.

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Step 3.10.1.1

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

$z = \frac{2 \pm \sqrt{4 - 4 \cdot 1 \cdot 24}}{2 \cdot 1}$

Step 3.10.1.2

Multiply $- 4 \cdot 1 \cdot 24$ .

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Step 3.10.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{2 \pm \sqrt{4 - 4 \cdot 24}}{2 \cdot 1}$

Step 3.10.1.2.2

Multiply $- 4$ by $24$ .

$z = \frac{2 \pm \sqrt{4 - 96}}{2 \cdot 1}$

Step 3.10.1.3

Subtract $96$ from $4$ .

$z = \frac{2 \pm \sqrt{- 92}}{2 \cdot 1}$

Step 3.10.1.4

Rewrite $- 92$ as $- 1 (92)$ .

$z = \frac{2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 1}$

Step 3.10.1.5

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$z = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 1}$

Step 3.10.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$z = \frac{2 \pm i \cdot \sqrt{92}}{2 \cdot 1}$

Step 3.10.1.7

Rewrite $92$ as $2^{2} \cdot 23$ .

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Step 3.10.1.7.1

Factor $4$ out of $92$ .

$z = \frac{2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 1}$

Step 3.10.1.7.2

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

$z = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 1}$

Step 3.10.1.8

Pull terms out from under the radical.

$z = \frac{2 \pm i \cdot (2 \sqrt{23})}{2 \cdot 1}$

Step 3.10.1.9

Move $2$ to the left of $i$ .

$z = \frac{2 \pm 2 i \sqrt{23}}{2 \cdot 1}$

Step 3.10.2

Multiply $2$ by $1$ .

$z = \frac{2 \pm 2 i \sqrt{23}}{2}$

Step 3.10.3

Simplify $\frac{2 \pm 2 i \sqrt{23}}{2}$ .

$z = 1 \pm i \sqrt{23}$

Step 3.11

The final answer is the combination of both solutions.

$z = 1 + i \sqrt{23}, 1 - i \sqrt{23}$