Solve for m 30m^2=216-m^4

$30 m^{2} = 216 - m^{4}$

Step 1

Add $m^{4}$ to both sides of the equation.

$30 m^{2} + m^{4} = 216$

Step 2

Subtract $216$ from both sides of the equation.

$30 m^{2} + m^{4} - 216 = 0$

Step 3

Factor $30 m^{2} + m^{4} - 216$ 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 $- 216$ and whose sum is $30$ .

$- 6, 36$

Write the factored form using these integers.

$(m^{2} - 6) (m^{2} + 36) = 0$

Step 4

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

$m^{2} - 6 = 0$

$m^{2} + 36 = 0$

Step 5

Set $m^{2} - 6$ equal to $0$ and solve for $m$ .

Set $m^{2} - 6$ equal to $0$ .

$m^{2} - 6 = 0$

Solve $m^{2} - 6 = 0$ for $m$ .

Add $6$ to both sides of the equation.

$m^{2} = 6$

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

$m = \pm \sqrt{6}$

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.

$m = \sqrt{6}$

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

$m = - \sqrt{6}$

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

$m = \sqrt{6}, - \sqrt{6}$

Step 6

Set $m^{2} + 36$ equal to $0$ and solve for $m$ .

Set $m^{2} + 36$ equal to $0$ .

$m^{2} + 36 = 0$

Solve $m^{2} + 36 = 0$ for $m$ .

Subtract $36$ from both sides of the equation.

$m^{2} = - 36$

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

$m = \pm \sqrt{- 36}$

Simplify $\pm \sqrt{- 36}$ .

Rewrite $- 36$ as $- 1 (36)$ .

$m = \pm \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$m = \pm \sqrt{- 1} \cdot \sqrt{36}$

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

$m = \pm i \cdot \sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$m = \pm i \cdot \sqrt{6^{2}}$

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

$m = \pm i \cdot 6$

Move $6$ to the left of $i$ .

$m = \pm 6 i$

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.

$m = 6 i$

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

$m = - 6 i$

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

$m = 6 i, - 6 i$

Step 7

The final solution is all the values that make $(m^{2} - 6) (m^{2} + 36) = 0$ true.

$m = \sqrt{6}, - \sqrt{6}, 6 i, - 6 i$

Solve for m 30m^2=216-m^4

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