Pre-Algebra Examples

$\frac{z^{2}}{4} = \frac{z}{5} + \frac{9}{20}$

Step 1

Multiply both sides by $4$ .

$\frac{z^{2}}{4} \cdot 4 = (\frac{z}{5} + \frac{9}{20}) \cdot 4$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $4$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{z^{2}}{4} \cdot 4 = (\frac{z}{5} + \frac{9}{20}) \cdot 4$

Step 2.1.1.2

Rewrite the expression.

$z^{2} = (\frac{z}{5} + \frac{9}{20}) \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $(\frac{z}{5} + \frac{9}{20}) \cdot 4$ .

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Step 2.2.1.1

Simplify terms.

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Step 2.2.1.1.1

Apply the distributive property.

$z^{2} = \frac{z}{5} \cdot 4 + \frac{9}{20} \cdot 4$

Step 2.2.1.1.2

Combine $\frac{z}{5}$ and $4$ .

$z^{2} = \frac{z \cdot 4}{5} + \frac{9}{20} \cdot 4$

Step 2.2.1.1.3

Cancel the common factor of $4$ .

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Step 2.2.1.1.3.1

Factor $4$ out of $20$ .

$z^{2} = \frac{z \cdot 4}{5} + \frac{9}{4 (5)} \cdot 4$

Step 2.2.1.1.3.2

Cancel the common factor.

$z^{2} = \frac{z \cdot 4}{5} + \frac{9}{4 \cdot 5} \cdot 4$

Step 2.2.1.1.3.3

Rewrite the expression.

$z^{2} = \frac{z \cdot 4}{5} + \frac{9}{5}$

Step 2.2.1.2

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

$z^{2} = \frac{4 z}{5} + \frac{9}{5}$

Step 3

Solve for $z$ .

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Step 3.1

Subtract $\frac{4 z}{5}$ from both sides of the equation.

$z^{2} - \frac{4 z}{5} = \frac{9}{5}$

Step 3.2

Subtract $\frac{9}{5}$ from both sides of the equation.

$z^{2} - \frac{4 z}{5} - \frac{9}{5} = 0$

Step 3.3

Multiply through by the least common denominator $5$ , then simplify.

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Step 3.3.1

Apply the distributive property.

$5 z^{2} + 5 (- \frac{4 z}{5}) + 5 (- \frac{9}{5}) = 0$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Move the leading negative in $- \frac{4 z}{5}$ into the numerator.

$5 z^{2} + 5 (\frac{- 4 z}{5}) + 5 (- \frac{9}{5}) = 0$

Step 3.3.2.1.2

Cancel the common factor.

$5 z^{2} + 5 (\frac{- 4 z}{5}) + 5 (- \frac{9}{5}) = 0$

Step 3.3.2.1.3

Rewrite the expression.

$5 z^{2} - 4 z + 5 (- \frac{9}{5}) = 0$

Step 3.3.2.2

Cancel the common factor of $5$ .

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Step 3.3.2.2.1

Move the leading negative in $- \frac{9}{5}$ into the numerator.

$5 z^{2} - 4 z + 5 (\frac{- 9}{5}) = 0$

Step 3.3.2.2.2

Cancel the common factor.

$5 z^{2} - 4 z + 5 (\frac{- 9}{5}) = 0$

Step 3.3.2.2.3

Rewrite the expression.

$5 z^{2} - 4 z - 9 = 0$

Step 3.4

Use the quadratic formula to find the solutions.

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

Step 3.5

Substitute the values $a = 5$ , $b = - 4$ , and $c = - 9$ into the quadratic formula and solve for $z$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (5 \cdot - 9)}}{2 \cdot 5}$

Step 3.6

Simplify.

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Step 3.6.1

Simplify the numerator.

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Step 3.6.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 5 \cdot - 9}}{2 \cdot 5}$

Step 3.6.1.2

Multiply $- 4 \cdot 5 \cdot - 9$ .

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Step 3.6.1.2.1

Multiply $- 4$ by $5$ .

$z = \frac{4 \pm \sqrt{16 - 20 \cdot - 9}}{2 \cdot 5}$

Step 3.6.1.2.2

Multiply $- 20$ by $- 9$ .

$z = \frac{4 \pm \sqrt{16 + 180}}{2 \cdot 5}$

Step 3.6.1.3

Add $16$ and $180$ .

$z = \frac{4 \pm \sqrt{196}}{2 \cdot 5}$

Step 3.6.1.4

Rewrite $196$ as $14^{2}$ .

$z = \frac{4 \pm \sqrt{14^{2}}}{2 \cdot 5}$

Step 3.6.1.5

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

$z = \frac{4 \pm 14}{2 \cdot 5}$

Step 3.6.2

Multiply $2$ by $5$ .

$z = \frac{4 \pm 14}{10}$

Step 3.6.3

Simplify $\frac{4 \pm 14}{10}$ .

$z = \frac{2 \pm 7}{5}$

Step 3.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.7.1

Simplify the numerator.

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Step 3.7.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 5 \cdot - 9}}{2 \cdot 5}$

Step 3.7.1.2

Multiply $- 4 \cdot 5 \cdot - 9$ .

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Step 3.7.1.2.1

Multiply $- 4$ by $5$ .

$z = \frac{4 \pm \sqrt{16 - 20 \cdot - 9}}{2 \cdot 5}$

Step 3.7.1.2.2

Multiply $- 20$ by $- 9$ .

$z = \frac{4 \pm \sqrt{16 + 180}}{2 \cdot 5}$

Step 3.7.1.3

Add $16$ and $180$ .

$z = \frac{4 \pm \sqrt{196}}{2 \cdot 5}$

Step 3.7.1.4

Rewrite $196$ as $14^{2}$ .

$z = \frac{4 \pm \sqrt{14^{2}}}{2 \cdot 5}$

Step 3.7.1.5

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

$z = \frac{4 \pm 14}{2 \cdot 5}$

Step 3.7.2

Multiply $2$ by $5$ .

$z = \frac{4 \pm 14}{10}$

Step 3.7.3

Simplify $\frac{4 \pm 14}{10}$ .

$z = \frac{2 \pm 7}{5}$

Step 3.7.4

Change the $\pm$ to $+$ .

$z = \frac{2 + 7}{5}$

Step 3.7.5

Add $2$ and $7$ .

$z = \frac{9}{5}$

Step 3.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.8.1

Simplify the numerator.

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Step 3.8.1.1

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

$z = \frac{4 \pm \sqrt{16 - 4 \cdot 5 \cdot - 9}}{2 \cdot 5}$

Step 3.8.1.2

Multiply $- 4 \cdot 5 \cdot - 9$ .

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Step 3.8.1.2.1

Multiply $- 4$ by $5$ .

$z = \frac{4 \pm \sqrt{16 - 20 \cdot - 9}}{2 \cdot 5}$

Step 3.8.1.2.2

Multiply $- 20$ by $- 9$ .

$z = \frac{4 \pm \sqrt{16 + 180}}{2 \cdot 5}$

Step 3.8.1.3

Add $16$ and $180$ .

$z = \frac{4 \pm \sqrt{196}}{2 \cdot 5}$

Step 3.8.1.4

Rewrite $196$ as $14^{2}$ .

$z = \frac{4 \pm \sqrt{14^{2}}}{2 \cdot 5}$

Step 3.8.1.5

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

$z = \frac{4 \pm 14}{2 \cdot 5}$

Step 3.8.2

Multiply $2$ by $5$ .

$z = \frac{4 \pm 14}{10}$

Step 3.8.3

Simplify $\frac{4 \pm 14}{10}$ .

$z = \frac{2 \pm 7}{5}$

Step 3.8.4

Change the $\pm$ to $-$ .

$z = \frac{2 - 7}{5}$

Step 3.8.5

Subtract $7$ from $2$ .

$z = \frac{- 5}{5}$

Step 3.8.6

Divide $- 5$ by $5$ .

$z = - 1$

Step 3.9

The final answer is the combination of both solutions.

$z = \frac{9}{5}, - 1$