Basic Math Examples

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

Step 1

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

$4 u^{2} - 5 u + 9 = 0$

$u = z^{2}$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 5$ to the power of $2$ .

$u = \frac{5 \pm \sqrt{25 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Step 4.1.2

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

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Step 4.1.2.1

Multiply $- 4$ by $4$ .

$u = \frac{5 \pm \sqrt{25 - 16 \cdot 9}}{2 \cdot 4}$

Step 4.1.2.2

Multiply $- 16$ by $9$ .

$u = \frac{5 \pm \sqrt{25 - 144}}{2 \cdot 4}$

Step 4.1.3

Subtract $144$ from $25$ .

$u = \frac{5 \pm \sqrt{- 119}}{2 \cdot 4}$

Step 4.1.4

Rewrite $- 119$ as $- 1 (119)$ .

$u = \frac{5 \pm \sqrt{- 1 \cdot 119}}{2 \cdot 4}$

Step 4.1.5

Rewrite $\sqrt{- 1 (119)}$ as $\sqrt{- 1} \cdot \sqrt{119}$ .

$u = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{119}}{2 \cdot 4}$

Step 4.1.6

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

$u = \frac{5 \pm i \sqrt{119}}{2 \cdot 4}$

Step 4.2

Multiply $2$ by $4$ .

$u = \frac{5 \pm i \sqrt{119}}{8}$

Step 5

The final answer is the combination of both solutions.

$u = \frac{5 + i \sqrt{119}}{8}, \frac{5 - i \sqrt{119}}{8}$

Step 6

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

$z^{2} = \frac{5 + i \sqrt{119}}{8}$

${(z^{2})}^{1} = \frac{5 - i \sqrt{119}}{8}$

Step 7

Solve the first equation for $z$ .

$z^{2} = \frac{5 + i \sqrt{119}}{8}$

Step 8

Solve the equation for $z$ .

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Step 8.1

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

$z = \pm \sqrt{\frac{5 + i \sqrt{119}}{8}}$

Step 8.2

Simplify $\pm \sqrt{\frac{5 + i \sqrt{119}}{8}}$ .

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Step 8.2.1

Rewrite $\sqrt{\frac{5 + i \sqrt{119}}{8}}$ as $\frac{\sqrt{5 + i \sqrt{119}}}{\sqrt{8}}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}}}{\sqrt{8}}$

Step 8.2.2

Simplify the denominator.

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Step 8.2.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 8.2.2.1.1

Factor $4$ out of $8$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}}}{\sqrt{4 (2)}}$

Step 8.2.2.1.2

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

$z = \pm \frac{\sqrt{5 + i \sqrt{119}}}{\sqrt{2^{2} \cdot 2}}$

Step 8.2.2.2

Pull terms out from under the radical.

$z = \pm \frac{\sqrt{5 + i \sqrt{119}}}{2 \sqrt{2}}$

Step 8.2.3

Multiply $\frac{\sqrt{5 + i \sqrt{119}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 8.2.4

Combine and simplify the denominator.

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Step 8.2.4.1

Multiply $\frac{\sqrt{5 + i \sqrt{119}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 8.2.4.2

Move $\sqrt{2}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.2.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 8.2.4.4

Raise $\sqrt{2}$ to the power of $1$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 8.2.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 8.2.4.6

Add $1$ and $1$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 8.2.4.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 8.2.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 8.2.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 8.2.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.2.4.7.4

Cancel the common factor of $2$ .

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Step 8.2.4.7.4.1

Cancel the common factor.

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.2.4.7.4.2

Rewrite the expression.

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{1}}$

Step 8.2.4.7.5

Evaluate the exponent.

$z = \pm \frac{\sqrt{5 + i \sqrt{119}} \sqrt{2}}{2 \cdot 2}$

Step 8.2.5

Combine using the product rule for radicals.

$z = \pm \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{2 \cdot 2}$

Step 8.2.6

Multiply $2$ by $2$ .

$z = \pm \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}$

Step 8.3

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

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Step 8.3.1

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

$z = \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}$

Step 8.3.2

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

$z = - \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}$

Step 8.3.3

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

$z = \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}$

Step 9

Solve the second equation for $z$ .

${(z^{2})}^{1} = \frac{5 - i \sqrt{119}}{8}$

Step 10

Solve the equation for $z$ .

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Step 10.1

Remove parentheses.

$z^{2} = \frac{5 - i \sqrt{119}}{8}$

Step 10.2

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

$z = \pm \sqrt{\frac{5 - i \sqrt{119}}{8}}$

Step 10.3

Simplify $\pm \sqrt{\frac{5 - i \sqrt{119}}{8}}$ .

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Step 10.3.1

Rewrite $\sqrt{\frac{5 - i \sqrt{119}}{8}}$ as $\frac{\sqrt{5 - i \sqrt{119}}}{\sqrt{8}}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}}}{\sqrt{8}}$

Step 10.3.2

Simplify the denominator.

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Step 10.3.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 10.3.2.1.1

Factor $4$ out of $8$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}}}{\sqrt{4 (2)}}$

Step 10.3.2.1.2

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

$z = \pm \frac{\sqrt{5 - i \sqrt{119}}}{\sqrt{2^{2} \cdot 2}}$

Step 10.3.2.2

Pull terms out from under the radical.

$z = \pm \frac{\sqrt{5 - i \sqrt{119}}}{2 \sqrt{2}}$

Step 10.3.3

Multiply $\frac{\sqrt{5 - i \sqrt{119}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 10.3.4

Combine and simplify the denominator.

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Step 10.3.4.1

Multiply $\frac{\sqrt{5 - i \sqrt{119}}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 10.3.4.2

Move $\sqrt{2}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 10.3.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 10.3.4.4

Raise $\sqrt{2}$ to the power of $1$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 10.3.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 10.3.4.6

Add $1$ and $1$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 10.3.4.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 10.3.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 10.3.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 10.3.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.4.7.4

Cancel the common factor of $2$ .

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Step 10.3.4.7.4.1

Cancel the common factor.

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 10.3.4.7.4.2

Rewrite the expression.

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \cdot 2^{1}}$

Step 10.3.4.7.5

Evaluate the exponent.

$z = \pm \frac{\sqrt{5 - i \sqrt{119}} \sqrt{2}}{2 \cdot 2}$

Step 10.3.5

Combine using the product rule for radicals.

$z = \pm \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{2 \cdot 2}$

Step 10.3.6

Multiply $2$ by $2$ .

$z = \pm \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$

Step 10.4

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

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Step 10.4.1

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

$z = \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$

Step 10.4.2

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

$z = - \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$

Step 10.4.3

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

$z = \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$

Step 11

The solution to $4 z^{4} - 5 z^{2} + 9 = 0$ is $z = \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}, \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$ .

$z = \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 + i \sqrt{119}) \cdot 2}}{4}, \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}, - \frac{\sqrt{(5 - i \sqrt{119}) \cdot 2}}{4}$