Solve by Factoring p-2 square root of p=8

$p - 2 \sqrt{p} = 8$

Step 1

Subtract $8$ from both sides of the equation.

$p - 2 \sqrt{p} - 8 = 0$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{p}$ as $p^{\frac{1}{2}}$ .

$p - 2 p^{\frac{1}{2}} - 8 = 0$

Step 3

Rewrite $p$ as ${(p^{\frac{1}{2}})}^{2}$ .

${(p^{\frac{1}{2}})}^{2} - 2 p^{\frac{1}{2}} - 8 = 0$

Step 4

Let $u = p^{\frac{1}{2}}$ . Substitute $u$ for all occurrences of $p^{\frac{1}{2}}$ .

$u^{2} - 2 u - 8 = 0$

Step 5

Factor $u^{2} - 2 u - 8$ 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 $- 8$ and whose sum is $- 2$ .

$- 4, 2$

Write the factored form using these integers.

$(u - 4) (u + 2) = 0$

Step 6

Replace all occurrences of $u$ with $p^{\frac{1}{2}}$ .

$(p^{\frac{1}{2}} - 4) (p^{\frac{1}{2}} + 2) = 0$

Step 7

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

$p^{\frac{1}{2}} - 4 = 0$

$p^{\frac{1}{2}} + 2 = 0$

Step 8

Set $p^{\frac{1}{2}} - 4$ equal to $0$ and solve for $p$ .

Set $p^{\frac{1}{2}} - 4$ equal to $0$ .

$p^{\frac{1}{2}} - 4 = 0$

Solve $p^{\frac{1}{2}} - 4 = 0$ for $p$ .

Add $4$ to both sides of the equation.

$p^{\frac{1}{2}} = 4$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(p^{\frac{1}{2}})}^{2} = 4^{2}$

Simplify the exponent.

Simplify the left side.

Simplify ${(p^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(p^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$p^{\frac{1}{2} \cdot 2} = 4^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$p^{\frac{1}{2} \cdot 2} = 4^{2}$

Rewrite the expression.

$p^{1} = 4^{2}$

Simplify.

$p = 4^{2}$

Simplify the right side.

Raise $4$ to the power of $2$ .

$p = 16$

Step 9

Set $p^{\frac{1}{2}} + 2$ equal to $0$ and solve for $p$ .

Set $p^{\frac{1}{2}} + 2$ equal to $0$ .

$p^{\frac{1}{2}} + 2 = 0$

Solve $p^{\frac{1}{2}} + 2 = 0$ for $p$ .

Subtract $2$ from both sides of the equation.

$p^{\frac{1}{2}} = - 2$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(p^{\frac{1}{2}})}^{2} = {(- 2)}^{2}$

Simplify the exponent.

Simplify the left side.

Simplify ${(p^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${(p^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$p^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$p^{\frac{1}{2} \cdot 2} = {(- 2)}^{2}$

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Raise $- 2$ to the power of $2$ .

$p = 4$

Step 10

The final solution is all the values that make $(p^{\frac{1}{2}} - 4) (p^{\frac{1}{2}} + 2) = 0$ true.

$p = 16, 4$

Step 11

Exclude the solutions that do not make $(p^{\frac{1}{2}} - 4) (p^{\frac{1}{2}} + 2) = 0$ true.

$p = 16$

Solve by Factoring p-2 square root of p=8

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