Solve by Factoring p^4-81=0

$p^{4} - 81 = 0$

Step 1

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

${(p^{2})}^{2} - 81 = 0$

Step 2

Rewrite $81$ as $9^{2}$ .

${(p^{2})}^{2} - 9^{2} = 0$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p^{2}$ and $b = 9$ .

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

Step 4

Simplify.

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

$(p^{2} + 9) (p^{2} - 3^{2}) = 0$

Factor.

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = p$ and $b = 3$ .

$(p^{2} + 9) ((p + 3) (p - 3)) = 0$

Remove unnecessary parentheses.

$(p^{2} + 9) (p + 3) (p - 3) = 0$

Step 5

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

$p^{2} + 9 = 0$

$p + 3 = 0$

$p - 3 = 0$

Step 6

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

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

$p^{2} + 9 = 0$

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

Subtract $9$ from both sides of the equation.

$p^{2} = - 9$

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

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

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

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

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

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

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

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

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

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

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

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

$p = \pm i \cdot 3$

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

$p = \pm 3 i$

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.

$p = 3 i$

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

$p = - 3 i$

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

$p = 3 i, - 3 i$

Step 7

Set $p + 3$ equal to $0$ and solve for $p$ .

Set $p + 3$ equal to $0$ .

$p + 3 = 0$

Subtract $3$ from both sides of the equation.

$p = - 3$

Step 8

Set $p - 3$ equal to $0$ and solve for $p$ .

Set $p - 3$ equal to $0$ .

$p - 3 = 0$

Add $3$ to both sides of the equation.

$p = 3$

Step 9

The final solution is all the values that make $(p^{2} + 9) (p + 3) (p - 3) = 0$ true.

$p = 3 i, - 3 i, - 3, 3$

Solve by Factoring p^4-81=0

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