Algebra Examples

$P (m) = (m^{2} - 4) (m^{2} + 1)$

Step 1

Set $(m^{2} - 4) (m^{2} + 1)$ equal to $0$ .

$(m^{2} - 4) (m^{2} + 1) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Simplify $(m^{2} - 4) (m^{2} + 1)$ .

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Step 2.1.1

Expand $(m^{2} - 4) (m^{2} + 1)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$m^{2} (m^{2} + 1) - 4 (m^{2} + 1) = 0$

Step 2.1.1.2

Apply the distributive property.

$m^{2} m^{2} + m^{2} \cdot 1 - 4 (m^{2} + 1) = 0$

Step 2.1.1.3

Apply the distributive property.

$m^{2} m^{2} + m^{2} \cdot 1 - 4 m^{2} - 4 \cdot 1 = 0$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Multiply $m^{2}$ by $m^{2}$ by adding the exponents.

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Step 2.1.2.1.1.1

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

$m^{2 + 2} + m^{2} \cdot 1 - 4 m^{2} - 4 \cdot 1 = 0$

Step 2.1.2.1.1.2

Add $2$ and $2$ .

$m^{4} + m^{2} \cdot 1 - 4 m^{2} - 4 \cdot 1 = 0$

Step 2.1.2.1.2

Multiply $m^{2}$ by $1$ .

$m^{4} + m^{2} - 4 m^{2} - 4 \cdot 1 = 0$

Step 2.1.2.1.3

Multiply $- 4$ by $1$ .

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

Step 2.1.2.2

Subtract $4 m^{2}$ from $m^{2}$ .

$m^{4} - 3 m^{2} - 4 = 0$

Step 2.2

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

$u^{2} - 3 u - 4 = 0$

$u = m^{2}$

Step 2.3

Factor $u^{2} - 3 u - 4$ using the AC method.

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Step 2.3.1

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 $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Step 2.3.2

Write the factored form using these integers.

$(u - 4) (u + 1) = 0$

Step 2.4

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

$u - 4 = 0$

$u + 1 = 0$

Step 2.5

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

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Step 2.5.1

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

$u - 4 = 0$

Step 2.5.2

Add $4$ to both sides of the equation.

$u = 4$

Step 2.6

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 2.6.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 2.6.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 2.7

The final solution is all the values that make $(u - 4) (u + 1) = 0$ true.

$u = 4, - 1$

Step 2.8

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

$m^{2} = 4$

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

Step 2.9

Solve the first equation for $m$ .

$m^{2} = 4$

Step 2.10

Solve the equation for $m$ .

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Step 2.10.1

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

$m = \pm \sqrt{4}$

Step 2.10.2

Simplify $\pm \sqrt{4}$ .

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Step 2.10.2.1

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

$m = \pm \sqrt{2^{2}}$

Step 2.10.2.2

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

$m = \pm 2$

Step 2.10.3

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

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Step 2.10.3.1

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

$m = 2$

Step 2.10.3.2

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

$m = - 2$

Step 2.10.3.3

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

$m = 2, - 2$

Step 2.11

Solve the second equation for $m$ .

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

Step 2.12

Solve the equation for $m$ .

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Step 2.12.1

Remove parentheses.

$m^{2} = - 1$

Step 2.12.2

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

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

Step 2.12.3

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

$m = \pm i$

Step 2.12.4

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

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Step 2.12.4.1

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

$m = i$

Step 2.12.4.2

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

$m = - i$

Step 2.12.4.3

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

$m = i, - i$

Step 2.13

The solution to $m^{4} - 3 m^{2} - 4 = 0$ is $m = 2, - 2, i, - i$ .

$m = 2, - 2, i, - i$

Step 3