Basic Math Examples

${(1 + z)}^{5} - {(1 - z)}^{5} = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

One to any power is one.

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

Step 1.2.2

One to any power is one.

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

Step 1.2.3

Multiply $5$ by $1$ .

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

Step 1.2.4

One to any power is one.

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

Step 1.2.5

Multiply $10$ by $1$ .

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

Step 1.2.6

One to any power is one.

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

Step 1.2.7

Multiply $10$ by $1$ .

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

Step 1.2.8

Multiply $5$ by $1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - {(1 - z)}^{5} = 0$

Step 1.3

Use the Binomial Theorem.

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

Step 1.4

Simplify each term.

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Step 1.4.1

One to any power is one.

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

Step 1.4.2

One to any power is one.

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

Step 1.4.3

Multiply $5$ by $1$ .

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

Step 1.4.4

Multiply $- 1$ by $5$ .

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

Step 1.4.5

One to any power is one.

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

Step 1.4.6

Multiply $10$ by $1$ .

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

Step 1.4.7

Apply the product rule to $- z$ .

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

Step 1.4.8

Raise $- 1$ to the power of $2$ .

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

Step 1.4.9

Multiply $z^{2}$ by $1$ .

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

Step 1.4.10

One to any power is one.

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

Step 1.4.11

Multiply $10$ by $1$ .

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

Step 1.4.12

Apply the product rule to $- z$ .

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

Step 1.4.13

Raise $- 1$ to the power of $3$ .

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

Step 1.4.14

Multiply $- 1$ by $10$ .

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

Step 1.4.15

Multiply $5$ by $1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 {(- z)}^{4} + {(- z)}^{5}) = 0$

Step 1.4.16

Apply the product rule to $- z$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 ({(- 1)}^{4} z^{4}) + {(- z)}^{5}) = 0$

Step 1.4.17

Raise $- 1$ to the power of $4$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 (1 z^{4}) + {(- z)}^{5}) = 0$

Step 1.4.18

Multiply $z^{4}$ by $1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 z^{4} + {(- z)}^{5}) = 0$

Step 1.4.19

Apply the product rule to $- z$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 z^{4} + {(- 1)}^{5} z^{5}) = 0$

Step 1.4.20

Raise $- 1$ to the power of $5$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - (1 - 5 z + 10 z^{2} - 10 z^{3} + 5 z^{4} - z^{5}) = 0$

Step 1.5

Apply the distributive property.

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

Step 1.6

Simplify.

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Step 1.6.1

Multiply $- 1$ by $1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 - (- 5 z) - (10 z^{2}) - (- 10 z^{3}) - (5 z^{4}) + z^{5} = 0$

Step 1.6.2

Multiply $- 5$ by $- 1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - (10 z^{2}) - (- 10 z^{3}) - (5 z^{4}) + z^{5} = 0$

Step 1.6.3

Multiply $10$ by $- 1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - 10 z^{2} - (- 10 z^{3}) - (5 z^{4}) + z^{5} = 0$

Step 1.6.4

Multiply $- 10$ by $- 1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - 10 z^{2} + 10 z^{3} - (5 z^{4}) + z^{5} = 0$

Step 1.6.5

Multiply $5$ by $- 1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - 10 z^{2} + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.6.6

Multiply $- - z^{5}$ .

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Step 1.6.6.1

Multiply $- 1$ by $- 1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - 10 z^{2} + 10 z^{3} - 5 z^{4} + 1 z^{5} = 0$

Step 1.6.6.2

Multiply $z^{5}$ by $1$ .

$1 + 5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} - 1 + 5 z - 10 z^{2} + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.7

Subtract $1$ from $1$ .

$5 z + 10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} + 0 + 5 z - 10 z^{2} + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.8

Add $5 z$ and $0$ .

$10 z^{2} + 10 z^{3} + 5 z^{4} + z^{5} + 5 z + 5 z - 10 z^{2} + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.9

Subtract $10 z^{2}$ from $10 z^{2}$ .

$10 z^{3} + 5 z^{4} + z^{5} + 5 z + 5 z + 0 + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.10

Add $10 z^{3}$ and $0$ .

$10 z^{3} + 5 z^{4} + z^{5} + 5 z + 5 z + 10 z^{3} - 5 z^{4} + z^{5} = 0$

Step 1.11

Add $10 z^{3}$ and $10 z^{3}$ .

$20 z^{3} + 5 z^{4} + z^{5} + 5 z + 5 z - 5 z^{4} + z^{5} = 0$

Step 1.12

Subtract $5 z^{4}$ from $5 z^{4}$ .

$20 z^{3} + z^{5} + 5 z + 5 z + 0 + z^{5} = 0$

Step 1.13

Add $20 z^{3}$ and $0$ .

$20 z^{3} + z^{5} + 5 z + 5 z + z^{5} = 0$

Step 1.14

Add $z^{5}$ and $z^{5}$ .

$20 z^{3} + 2 z^{5} + 5 z + 5 z = 0$

Step 1.15

Add $5 z$ and $5 z$ .

$20 z^{3} + 2 z^{5} + 10 z = 0$

Step 1.16

Reorder terms.

$2 z^{5} + 20 z^{3} + 10 z = 0$

Step 1.17

Factor $2 z$ out of $2 z^{5} + 20 z^{3} + 10 z$ .

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Step 1.17.1

Factor $2 z$ out of $2 z^{5}$ .

$2 z (z^{4}) + 20 z^{3} + 10 z = 0$

Step 1.17.2

Factor $2 z$ out of $20 z^{3}$ .

$2 z (z^{4}) + 2 z (10 z^{2}) + 10 z = 0$

Step 1.17.3

Factor $2 z$ out of $10 z$ .

$2 z (z^{4}) + 2 z (10 z^{2}) + 2 z (5) = 0$

Step 1.17.4

Factor $2 z$ out of $2 z (z^{4}) + 2 z (10 z^{2})$ .

$2 z (z^{4} + 10 z^{2}) + 2 z (5) = 0$

Step 1.17.5

Factor $2 z$ out of $2 z (z^{4} + 10 z^{2}) + 2 z (5)$ .

$2 z (z^{4} + 10 z^{2} + 5) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z = 0$

$z^{4} + 10 z^{2} + 5 = 0$

Step 3

Set $z$ equal to $0$ .

$z = 0$

Step 4

Set $z^{4} + 10 z^{2} + 5$ equal to $0$ and solve for $z$ .

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Step 4.1

Set $z^{4} + 10 z^{2} + 5$ equal to $0$ .

$z^{4} + 10 z^{2} + 5 = 0$

Step 4.2

Solve $z^{4} + 10 z^{2} + 5 = 0$ for $z$ .

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Step 4.2.1

Substitute $u = z^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} + 10 u + 5 = 0$

$u = z^{2}$

Step 4.2.2

Use the quadratic formula to find the solutions.

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

Step 4.2.3

Substitute the values $a = 1$ , $b = 10$ , and $c = 5$ into the quadratic formula and solve for $u$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot 5)}}{2 \cdot 1}$

Step 4.2.4

Simplify.

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Step 4.2.4.1

Simplify the numerator.

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Step 4.2.4.1.1

Raise $10$ to the power of $2$ .

$u = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$

Step 4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 5$ .

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Step 4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{- 10 \pm \sqrt{100 - 4 \cdot 5}}{2 \cdot 1}$

Step 4.2.4.1.2.2

Multiply $- 4$ by $5$ .

$u = \frac{- 10 \pm \sqrt{100 - 20}}{2 \cdot 1}$

Step 4.2.4.1.3

Subtract $20$ from $100$ .

$u = \frac{- 10 \pm \sqrt{80}}{2 \cdot 1}$

Step 4.2.4.1.4

Rewrite $80$ as $4^{2} \cdot 5$ .

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Step 4.2.4.1.4.1

Factor $16$ out of $80$ .

$u = \frac{- 10 \pm \sqrt{16 (5)}}{2 \cdot 1}$

Step 4.2.4.1.4.2

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

$u = \frac{- 10 \pm \sqrt{4^{2} \cdot 5}}{2 \cdot 1}$

Step 4.2.4.1.5

Pull terms out from under the radical.

$u = \frac{- 10 \pm 4 \sqrt{5}}{2 \cdot 1}$

Step 4.2.4.2

Multiply $2$ by $1$ .

$u = \frac{- 10 \pm 4 \sqrt{5}}{2}$

Step 4.2.4.3

Simplify $\frac{- 10 \pm 4 \sqrt{5}}{2}$ .

$u = - 5 \pm 2 \sqrt{5}$

Step 4.2.5

The final answer is the combination of both solutions.

$u = - 5 + 2 \sqrt{5}, - 5 - 2 \sqrt{5}$

Step 4.2.6

Substitute the real value of $u = z^{2}$ back into the solved equation.

$z^{2} = - 0.52786404$

${(z^{2})}^{1} = - 9.47213595$

Step 4.2.7

Solve the first equation for $z$ .

$z^{2} = - 0.52786404$

Step 4.2.8

Solve the equation for $z$ .

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Step 4.2.8.1

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

$z = \pm \sqrt{- 0.52786404}$

Step 4.2.8.2

Simplify $\pm \sqrt{- 0.52786404}$ .

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Step 4.2.8.2.1

Rewrite $- 0.52786404$ as $- 1 (0.52786404)$ .

$z = \pm \sqrt{- 1 (0.52786404)}$

Step 4.2.8.2.2

Rewrite $\sqrt{- 1 (0.52786404)}$ as $\sqrt{- 1} \cdot \sqrt{0.52786404}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{0.52786404}$

Step 4.2.8.2.3

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

$z = \pm i \sqrt{0.52786404}$

Step 4.2.8.3

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

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Step 4.2.8.3.1

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

$z = i \sqrt{0.52786404}$

Step 4.2.8.3.2

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

$z = - i \sqrt{0.52786404}$

Step 4.2.8.3.3

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

$z = i \sqrt{0.52786404}, - i \sqrt{0.52786404}$

Step 4.2.9

Solve the second equation for $z$ .

${(z^{2})}^{1} = - 9.47213595$

Step 4.2.10

Solve the equation for $z$ .

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Step 4.2.10.1

Remove parentheses.

$z^{2} = - 9.47213595$

Step 4.2.10.2

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

$z = \pm \sqrt{- 9.47213595}$

Step 4.2.10.3

Simplify $\pm \sqrt{- 9.47213595}$ .

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Step 4.2.10.3.1

Rewrite $- 9.47213595$ as $- 1 (9.47213595)$ .

$z = \pm \sqrt{- 1 (9.47213595)}$

Step 4.2.10.3.2

Rewrite $\sqrt{- 1 (9.47213595)}$ as $\sqrt{- 1} \cdot \sqrt{9.47213595}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{9.47213595}$

Step 4.2.10.3.3

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

$z = \pm i \sqrt{9.47213595}$

Step 4.2.10.4

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

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Step 4.2.10.4.1

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

$z = i \sqrt{9.47213595}$

Step 4.2.10.4.2

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

$z = - i \sqrt{9.47213595}$

Step 4.2.10.4.3

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

$z = i \sqrt{9.47213595}, - i \sqrt{9.47213595}$

Step 4.2.11

The solution to $z^{4} + 10 z^{2} + 5 = 0$ is $z = i \sqrt{0.52786404}, - i \sqrt{0.52786404}, i \sqrt{9.47213595}, - i \sqrt{9.47213595}$ .

$z = i \sqrt{0.52786404}, - i \sqrt{0.52786404}, i \sqrt{9.47213595}, - i \sqrt{9.47213595}$

Step 5

The final solution is all the values that make $2 z (z^{4} + 10 z^{2} + 5) = 0$ true.

$z = 0, i \sqrt{0.52786404}, - i \sqrt{0.52786404}, i \sqrt{9.47213595}, - i \sqrt{9.47213595}$