Pre-Algebra Examples

$- (5 z (10 z + 2)) = 2 + (4 z + 3)$

Step 1

Simplify $- 1 \cdot (5 z (10 z + 2))$ .

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Step 1.1

Rewrite.

$0 + 0 - 1 \cdot (5 z (10 z + 2)) = 2 + 4 z + 3$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$- 1 \cdot (5 z (10 z) + 5 z \cdot 2) = 2 + 4 z + 3$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$- 1 \cdot (5 \cdot 10 z \cdot z + 5 z \cdot 2) = 2 + 4 z + 3$

Step 1.2.2.2

Multiply $2$ by $5$ .

$- 1 \cdot (5 \cdot 10 z \cdot z + 10 z) = 2 + 4 z + 3$

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Move $z$ .

$- 1 \cdot (5 \cdot 10 (z \cdot z) + 10 z) = 2 + 4 z + 3$

Step 1.3.1.2

Multiply $z$ by $z$ .

$- 1 \cdot (5 \cdot 10 z^{2} + 10 z) = 2 + 4 z + 3$

Step 1.3.2

Multiply $5$ by $10$ .

$- 1 \cdot (50 z^{2} + 10 z) = 2 + 4 z + 3$

Step 1.4

Simplify by multiplying through.

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Step 1.4.1

Apply the distributive property.

$- 1 (50 z^{2}) - 1 (10 z) = 2 + 4 z + 3$

Step 1.4.2

Multiply.

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Step 1.4.2.1

Multiply $50$ by $- 1$ .

$- 50 z^{2} - 1 (10 z) = 2 + 4 z + 3$

Step 1.4.2.2

Multiply $10$ by $- 1$ .

$- 50 z^{2} - 10 z = 2 + 4 z + 3$

Step 2

Add $2$ and $3$ .

$- 50 z^{2} - 10 z = 4 z + 5$

Step 3

Move all terms containing $z$ to the left side of the equation.

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Step 3.1

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

$- 50 z^{2} - 10 z - 4 z = 5$

Step 3.2

Subtract $4 z$ from $- 10 z$ .

$- 50 z^{2} - 14 z = 5$

Step 4

Subtract $5$ from both sides of the equation.

$- 50 z^{2} - 14 z - 5 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 50$ , $b = - 14$ , and $c = - 5$ into the quadratic formula and solve for $z$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (- 50 \cdot - 5)}}{2 \cdot - 50}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 14$ to the power of $2$ .

$z = \frac{14 \pm \sqrt{196 - 4 \cdot - 50 \cdot - 5}}{2 \cdot - 50}$

Step 7.1.2

Multiply $- 4 \cdot - 50 \cdot - 5$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 50$ .

$z = \frac{14 \pm \sqrt{196 + 200 \cdot - 5}}{2 \cdot - 50}$

Step 7.1.2.2

Multiply $200$ by $- 5$ .

$z = \frac{14 \pm \sqrt{196 - 1000}}{2 \cdot - 50}$

Step 7.1.3

Subtract $1000$ from $196$ .

$z = \frac{14 \pm \sqrt{- 804}}{2 \cdot - 50}$

Step 7.1.4

Rewrite $- 804$ as $- 1 (804)$ .

$z = \frac{14 \pm \sqrt{- 1 \cdot 804}}{2 \cdot - 50}$

Step 7.1.5

Rewrite $\sqrt{- 1 (804)}$ as $\sqrt{- 1} \cdot \sqrt{804}$ .

$z = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{804}}{2 \cdot - 50}$

Step 7.1.6

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

$z = \frac{14 \pm i \cdot \sqrt{804}}{2 \cdot - 50}$

Step 7.1.7

Rewrite $804$ as $2^{2} \cdot 201$ .

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Step 7.1.7.1

Factor $4$ out of $804$ .

$z = \frac{14 \pm i \cdot \sqrt{4 (201)}}{2 \cdot - 50}$

Step 7.1.7.2

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

$z = \frac{14 \pm i \cdot \sqrt{2^{2} \cdot 201}}{2 \cdot - 50}$

Step 7.1.8

Pull terms out from under the radical.

$z = \frac{14 \pm i \cdot (2 \sqrt{201})}{2 \cdot - 50}$

Step 7.1.9

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

$z = \frac{14 \pm 2 i \sqrt{201}}{2 \cdot - 50}$

Step 7.2

Multiply $2$ by $- 50$ .

$z = \frac{14 \pm 2 i \sqrt{201}}{- 100}$

Step 7.3

Simplify $\frac{14 \pm 2 i \sqrt{201}}{- 100}$ .

$z = \frac{7 \pm i \sqrt{201}}{- 50}$

Step 7.4

Move the negative in front of the fraction.

$z = - \frac{7 \pm i \sqrt{201}}{50}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 14$ to the power of $2$ .

$z = \frac{14 \pm \sqrt{196 - 4 \cdot - 50 \cdot - 5}}{2 \cdot - 50}$

Step 8.1.2

Multiply $- 4 \cdot - 50 \cdot - 5$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 50$ .

$z = \frac{14 \pm \sqrt{196 + 200 \cdot - 5}}{2 \cdot - 50}$

Step 8.1.2.2

Multiply $200$ by $- 5$ .

$z = \frac{14 \pm \sqrt{196 - 1000}}{2 \cdot - 50}$

Step 8.1.3

Subtract $1000$ from $196$ .

$z = \frac{14 \pm \sqrt{- 804}}{2 \cdot - 50}$

Step 8.1.4

Rewrite $- 804$ as $- 1 (804)$ .

$z = \frac{14 \pm \sqrt{- 1 \cdot 804}}{2 \cdot - 50}$

Step 8.1.5

Rewrite $\sqrt{- 1 (804)}$ as $\sqrt{- 1} \cdot \sqrt{804}$ .

$z = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{804}}{2 \cdot - 50}$

Step 8.1.6

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

$z = \frac{14 \pm i \cdot \sqrt{804}}{2 \cdot - 50}$

Step 8.1.7

Rewrite $804$ as $2^{2} \cdot 201$ .

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Step 8.1.7.1

Factor $4$ out of $804$ .

$z = \frac{14 \pm i \cdot \sqrt{4 (201)}}{2 \cdot - 50}$

Step 8.1.7.2

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

$z = \frac{14 \pm i \cdot \sqrt{2^{2} \cdot 201}}{2 \cdot - 50}$

Step 8.1.8

Pull terms out from under the radical.

$z = \frac{14 \pm i \cdot (2 \sqrt{201})}{2 \cdot - 50}$

Step 8.1.9

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

$z = \frac{14 \pm 2 i \sqrt{201}}{2 \cdot - 50}$

Step 8.2

Multiply $2$ by $- 50$ .

$z = \frac{14 \pm 2 i \sqrt{201}}{- 100}$

Step 8.3

Simplify $\frac{14 \pm 2 i \sqrt{201}}{- 100}$ .

$z = \frac{7 \pm i \sqrt{201}}{- 50}$

Step 8.4

Move the negative in front of the fraction.

$z = - \frac{7 \pm i \sqrt{201}}{50}$

Step 8.5

Change the $\pm$ to $+$ .

$z = - \frac{7 + i \sqrt{201}}{50}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 14$ to the power of $2$ .

$z = \frac{14 \pm \sqrt{196 - 4 \cdot - 50 \cdot - 5}}{2 \cdot - 50}$

Step 9.1.2

Multiply $- 4 \cdot - 50 \cdot - 5$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 50$ .

$z = \frac{14 \pm \sqrt{196 + 200 \cdot - 5}}{2 \cdot - 50}$

Step 9.1.2.2

Multiply $200$ by $- 5$ .

$z = \frac{14 \pm \sqrt{196 - 1000}}{2 \cdot - 50}$

Step 9.1.3

Subtract $1000$ from $196$ .

$z = \frac{14 \pm \sqrt{- 804}}{2 \cdot - 50}$

Step 9.1.4

Rewrite $- 804$ as $- 1 (804)$ .

$z = \frac{14 \pm \sqrt{- 1 \cdot 804}}{2 \cdot - 50}$

Step 9.1.5

Rewrite $\sqrt{- 1 (804)}$ as $\sqrt{- 1} \cdot \sqrt{804}$ .

$z = \frac{14 \pm \sqrt{- 1} \cdot \sqrt{804}}{2 \cdot - 50}$

Step 9.1.6

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

$z = \frac{14 \pm i \cdot \sqrt{804}}{2 \cdot - 50}$

Step 9.1.7

Rewrite $804$ as $2^{2} \cdot 201$ .

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Step 9.1.7.1

Factor $4$ out of $804$ .

$z = \frac{14 \pm i \cdot \sqrt{4 (201)}}{2 \cdot - 50}$

Step 9.1.7.2

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

$z = \frac{14 \pm i \cdot \sqrt{2^{2} \cdot 201}}{2 \cdot - 50}$

Step 9.1.8

Pull terms out from under the radical.

$z = \frac{14 \pm i \cdot (2 \sqrt{201})}{2 \cdot - 50}$

Step 9.1.9

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

$z = \frac{14 \pm 2 i \sqrt{201}}{2 \cdot - 50}$

Step 9.2

Multiply $2$ by $- 50$ .

$z = \frac{14 \pm 2 i \sqrt{201}}{- 100}$

Step 9.3

Simplify $\frac{14 \pm 2 i \sqrt{201}}{- 100}$ .

$z = \frac{7 \pm i \sqrt{201}}{- 50}$

Step 9.4

Move the negative in front of the fraction.

$z = - \frac{7 \pm i \sqrt{201}}{50}$

Step 9.5

Change the $\pm$ to $-$ .

$z = - \frac{7 - i \sqrt{201}}{50}$

Step 10

The final answer is the combination of both solutions.

$z = - \frac{7 + i \sqrt{201}}{50}, - \frac{7 - i \sqrt{201}}{50}$