Solve by Factoring z^4+5z^2-36=0

$z^{4} + 5 z^{2} - 36 = 0$

Step 1

Rewrite $z^{4}$ as ${(z^{2})}^{2}$ .

${(z^{2})}^{2} + 5 z^{2} - 36 = 0$

Step 2

Let $u = z^{2}$ . Substitute $u$ for all occurrences of $z^{2}$ .

$u^{2} + 5 u - 36 = 0$

Step 3

Factor $u^{2} + 5 u - 36$ using the AC method.

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 $- 36$ and whose sum is $5$ .

$- 4, 9$

Write the factored form using these integers.

$(u - 4) (u + 9) = 0$

Step 4

Replace all occurrences of $u$ with $z^{2}$ .

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

Step 5

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

$(z^{2} - 2^{2}) (z^{2} + 9) = 0$

Step 6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = z$ and $b = 2$ .

$(z + 2) (z - 2) (z^{2} + 9) = 0$

Step 7

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

$z + 2 = 0$

$z - 2 = 0$

$z^{2} + 9 = 0$

Step 8

Set $z + 2$ equal to $0$ and solve for $z$ .

Set $z + 2$ equal to $0$ .

$z + 2 = 0$

Subtract $2$ from both sides of the equation.

$z = - 2$

Step 9

Set $z - 2$ equal to $0$ and solve for $z$ .

Set $z - 2$ equal to $0$ .

$z - 2 = 0$

Add $2$ to both sides of the equation.

$z = 2$

Step 10

Set $z^{2} + 9$ equal to $0$ and solve for $z$ .

Set $z^{2} + 9$ equal to $0$ .

$z^{2} + 9 = 0$

Solve $z^{2} + 9 = 0$ for $z$ .

Subtract $9$ from both sides of the equation.

$z^{2} = - 9$

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

$z = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$z = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$z = \pm i \cdot \sqrt{9}$

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

$z = \pm i \cdot \sqrt{3^{2}}$

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

$z = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$z = \pm 3 i$

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

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

$z = 3 i$

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

$z = - 3 i$

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

$z = 3 i, - 3 i$

Step 11

The final solution is all the values that make $(z + 2) (z - 2) (z^{2} + 9) = 0$ true.

$z = - 2, 2, 3 i, - 3 i$

Solve by Factoring z^4+5z^2-36=0

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