Solve by Factoring p^4-p^2-20=0

$p^{4} - p^{2} - 20 = 0$

Step 1

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

${(p^{2})}^{2} - p^{2} - 20 = 0$

Step 2

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

$u^{2} - u - 20 = 0$

Step 3

Factor $u^{2} - u - 20$ 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 $- 20$ and whose sum is $- 1$ .

$- 5, 4$

Write the factored form using these integers.

$(u - 5) (u + 4) = 0$

Step 4

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

$(p^{2} - 5) (p^{2} + 4) = 0$

Step 5

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

$p^{2} - 5 = 0$

$p^{2} + 4 = 0$

Step 6

Set $p^{2} - 5$ equal to $0$ and solve for $p$ .

Set $p^{2} - 5$ equal to $0$ .

$p^{2} - 5 = 0$

Solve $p^{2} - 5 = 0$ for $p$ .

Add $5$ to both sides of the equation.

$p^{2} = 5$

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

$p = \pm \sqrt{5}$

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.

$p = \sqrt{5}$

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

$p = - \sqrt{5}$

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

$p = \sqrt{5}, - \sqrt{5}$

Step 7

Set $p^{2} + 4$ equal to $0$ and solve for $p$ .

Set $p^{2} + 4$ equal to $0$ .

$p^{2} + 4 = 0$

Solve $p^{2} + 4 = 0$ for $p$ .

Subtract $4$ from both sides of the equation.

$p^{2} = - 4$

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

$p = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$p = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$p = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$p = \pm i \cdot \sqrt{4}$

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

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

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

$p = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$p = \pm 2 i$

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.

$p = 2 i$

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

$p = - 2 i$

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

$p = 2 i, - 2 i$

Step 8

The final solution is all the values that make $(p^{2} - 5) (p^{2} + 4) = 0$ true.

$p = \sqrt{5}, - \sqrt{5}, 2 i, - 2 i$

Solve by Factoring p^4-p^2-20=0

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