Algebra Examples

$25 m^{2} = 396 - m^{4}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $396$ from both sides of the equation.

$25 m^{2} - 396 = - m^{4}$

Step 1.2

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

$25 m^{2} - 396 + m^{4} = 0$

Step 2

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

$u^{2} + 25 u - 396 = 0$

$u = m^{2}$

Step 3

Factor $u^{2} + 25 u - 396$ using the AC method.

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Step 3.1

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 $- 396$ and whose sum is $25$ .

$- 11, 36$

Step 3.2

Write the factored form using these integers.

$(u - 11) (u + 36) = 0$

Step 4

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

$u - 11 = 0$

$u + 36 = 0$

Step 5

Set $u - 11$ equal to $0$ and solve for $u$ .

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Step 5.1

Set $u - 11$ equal to $0$ .

$u - 11 = 0$

Step 5.2

Add $11$ to both sides of the equation.

$u = 11$

Step 6

Set $u + 36$ equal to $0$ and solve for $u$ .

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Step 6.1

Set $u + 36$ equal to $0$ .

$u + 36 = 0$

Step 6.2

Subtract $36$ from both sides of the equation.

$u = - 36$

Step 7

The final solution is all the values that make $(u - 11) (u + 36) = 0$ true.

$u = 11, - 36$

Step 8

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

$m^{2} = 11$

${(m^{2})}^{1} = - 36$

Step 9

Solve the first equation for $m$ .

$m^{2} = 11$

Step 10

Solve the equation for $m$ .

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Step 10.1

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

$m = \pm \sqrt{11}$

Step 10.2

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

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Step 10.2.1

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

$m = \sqrt{11}$

Step 10.2.2

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

$m = - \sqrt{11}$

Step 10.2.3

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

$m = \sqrt{11}, - \sqrt{11}$

Step 11

Solve the second equation for $m$ .

${(m^{2})}^{1} = - 36$

Step 12

Solve the equation for $m$ .

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Step 12.1

Remove parentheses.

$m^{2} = - 36$

Step 12.2

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

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

Step 12.3

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

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Step 12.3.1

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

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

Step 12.3.2

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

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

Step 12.3.3

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

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

Step 12.3.4

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

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

Step 12.3.5

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

$m = \pm i \cdot 6$

Step 12.3.6

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

$m = \pm 6 i$

Step 12.4

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

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Step 12.4.1

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

$m = 6 i$

Step 12.4.2

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

$m = - 6 i$

Step 12.4.3

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

$m = 6 i, - 6 i$

Step 13

The solution to $m^{4} + 25 m^{2} - 396 = 0$ is $m = \sqrt{11}, - \sqrt{11}, 6 i, - 6 i$ .

$m = \sqrt{11}, - \sqrt{11}, 6 i, - 6 i$