Solve by Factoring p+p^(1/2)-2=0

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

Step 1

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

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

Step 2

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

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

Step 3

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

$- 1, 2$

Write the factored form using these integers.

$(u - 1) (u + 2) = 0$

Step 4

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

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

Step 5

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

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

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

Step 6

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

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

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

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

Add $1$ to both sides of the equation.

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

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$p = 1^{2}$

Simplify the right side.

One to any power is one.

$p = 1$

Step 7

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 8

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

$p = 1, 4$

Step 9

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

$p = 1$

Solve by Factoring p+p^(1/2)-2=0

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