Linear Algebra Examples

$z^{4} - 21 z^{2} - 100 = 0$

Step 1

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

$u^{2} - 21 u - 100 = 0$

$u = z^{2}$

Step 2

Factor $u^{2} - 21 u - 100$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 100$ and whose sum is $- 21$ .

$- 25, 4$

Step 2.2

Write the factored form using these integers.

$(u - 25) (u + 4) = 0$

Step 3

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

$u - 25 = 0$

$u + 4 = 0$

Step 4

Set $u - 25$ equal to $0$ and solve for $u$ .

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Step 4.1

Set $u - 25$ equal to $0$ .

$u - 25 = 0$

Step 4.2

Add $25$ to both sides of the equation.

$u = 25$

Step 5

Set $u + 4$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u + 4$ equal to $0$ .

$u + 4 = 0$

Step 5.2

Subtract $4$ from both sides of the equation.

$u = - 4$

Step 6

The final solution is all the values that make $(u - 25) (u + 4) = 0$ true.

$u = 25, - 4$

Step 7

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

$z^{2} = 25$

${(z^{2})}^{1} = - 4$

Step 8

Solve the first equation for $z$ .

$z^{2} = 25$

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.

$z = \pm \sqrt{25}$

Step 9.2

Simplify $\pm \sqrt{25}$ .

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Step 9.2.1

Rewrite $25$ as $5^{2}$ .

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

Step 9.2.2

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

$z = \pm 5$

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.

$z = 5$

Step 9.3.2

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

$z = - 5$

Step 9.3.3

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

$z = 5, - 5$

Step 10

Solve the second equation for $z$ .

${(z^{2})}^{1} = - 4$

Step 11

Solve the equation for $z$ .

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Step 11.1

Remove parentheses.

$z^{2} = - 4$

Step 11.2

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

$z = \pm \sqrt{- 4}$

Step 11.3

Simplify $\pm \sqrt{- 4}$ .

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Step 11.3.1

Rewrite $- 4$ as $- 1 (4)$ .

$z = \pm \sqrt{- 1 (4)}$

Step 11.3.2

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{4}$

Step 11.3.3

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

$z = \pm i \cdot \sqrt{4}$

Step 11.3.4

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

$z = \pm i \cdot \sqrt{2^{2}}$

Step 11.3.5

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

$z = \pm i \cdot 2$

Step 11.3.6

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

$z = \pm 2 i$

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.

$z = 2 i$

Step 11.4.2

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

$z = - 2 i$

Step 11.4.3

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

$z = 2 i, - 2 i$

Step 12

The solution to $z^{4} - 21 z^{2} - 100 = 0$ is $z = 5, - 5, 2 i, - 2 i$ .

$z = 5, - 5, 2 i, - 2 i$

Step 13

The domain is the set of all valid $z$ values.

${z | z = 5, - 5, 2 i, - 2 i}$

Step 14