Basic Math Examples

$5 z^{4} + 16 z^{2} - 45 = 0$

Step 1

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

$5 u^{2} + 16 u - 45 = 0$

$u = z^{2}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 45 = - 225$ and whose sum is $b = 16$ .

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Step 2.1.1

Factor $16$ out of $16 u$ .

$5 u^{2} + 16 (u) - 45 = 0$

Step 2.1.2

Rewrite $16$ as $- 9$ plus $25$

$5 u^{2} + (- 9 + 25) u - 45 = 0$

Step 2.1.3

Apply the distributive property.

$5 u^{2} - 9 u + 25 u - 45 = 0$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$(5 u^{2} - 9 u) + 25 u - 45 = 0$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$u (5 u - 9) + 5 (5 u - 9) = 0$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 9$ .

$(5 u - 9) (u + 5) = 0$

Step 3

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

$5 u - 9 = 0$

$u + 5 = 0$

Step 4

Set $5 u - 9$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $5 u - 9$ equal to $0$ .

$5 u - 9 = 0$

Step 4.2

Solve $5 u - 9 = 0$ for $u$ .

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Step 4.2.1

Add $9$ to both sides of the equation.

$5 u = 9$

Step 4.2.2

Divide each term in $5 u = 9$ by $5$ and simplify.

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Step 4.2.2.1

Divide each term in $5 u = 9$ by $5$ .

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

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $5$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.2.1.2

Divide $u$ by $1$ .

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

Step 5

Set $u + 5$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u + 5$ equal to $0$ .

$u + 5 = 0$

Step 5.2

Subtract $5$ from both sides of the equation.

$u = - 5$

Step 6

The final solution is all the values that make $(5 u - 9) (u + 5) = 0$ true.

$u = \frac{9}{5}, - 5$

Step 7

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

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

${(z^{2})}^{1} = - 5$

Step 8

Solve the first equation for $z$ .

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

Step 9

Solve the equation for $z$ .

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Step 9.1

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

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

Step 9.2

Simplify $\pm \sqrt{\frac{9}{5}}$ .

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Step 9.2.1

Rewrite $\sqrt{\frac{9}{5}}$ as $\frac{\sqrt{9}}{\sqrt{5}}$ .

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

Step 9.2.2

Simplify the numerator.

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Step 9.2.2.1

Rewrite $9$ as $3^{2}$ .

$z = \pm \frac{\sqrt{3^{2}}}{\sqrt{5}}$

Step 9.2.2.2

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

$z = \pm \frac{3}{\sqrt{5}}$

Step 9.2.3

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$z = \pm \frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 9.2.4

Combine and simplify the denominator.

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Step 9.2.4.1

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$z = \pm \frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 9.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$z = \pm \frac{3 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 9.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$z = \pm \frac{3 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 9.2.4.4

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

$z = \pm \frac{3 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 9.2.4.5

Add $1$ and $1$ .

$z = \pm \frac{3 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 9.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 9.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$z = \pm \frac{3 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 9.2.4.6.2

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

$z = \pm \frac{3 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 9.2.4.6.3

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

$z = \pm \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.2.4.6.4

Cancel the common factor of $2$ .

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Step 9.2.4.6.4.1

Cancel the common factor.

$z = \pm \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 9.2.4.6.4.2

Rewrite the expression.

$z = \pm \frac{3 \sqrt{5}}{5^{1}}$

Step 9.2.4.6.5

Evaluate the exponent.

$z = \pm \frac{3 \sqrt{5}}{5}$

Step 9.3

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

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Step 9.3.1

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

$z = \frac{3 \sqrt{5}}{5}$

Step 9.3.2

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

$z = - \frac{3 \sqrt{5}}{5}$

Step 9.3.3

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

$z = \frac{3 \sqrt{5}}{5}, - \frac{3 \sqrt{5}}{5}$

Step 10

Solve the second equation for $z$ .

${(z^{2})}^{1} = - 5$

Step 11

Solve the equation for $z$ .

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Step 11.1

Remove parentheses.

$z^{2} = - 5$

Step 11.2

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

$z = \pm \sqrt{- 5}$

Step 11.3

Simplify $\pm \sqrt{- 5}$ .

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Step 11.3.1

Rewrite $- 5$ as $- 1 (5)$ .

$z = \pm \sqrt{- 1 (5)}$

Step 11.3.2

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{5}$

Step 11.3.3

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

$z = \pm i \sqrt{5}$

Step 11.4

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

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Step 11.4.1

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

$z = i \sqrt{5}$

Step 11.4.2

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

$z = - i \sqrt{5}$

Step 11.4.3

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

$z = i \sqrt{5}, - i \sqrt{5}$

Step 12

The solution to $5 z^{4} + 16 z^{2} - 45 = 0$ is $z = \frac{3 \sqrt{5}}{5}, - \frac{3 \sqrt{5}}{5}, i \sqrt{5}, - i \sqrt{5}$ .

$z = \frac{3 \sqrt{5}}{5}, - \frac{3 \sqrt{5}}{5}, i \sqrt{5}, - i \sqrt{5}$