Solve for p square root of 11-p- square root of 2+p=-1

$\sqrt{11 - p} - \sqrt{2 + p} = - 1$

Add $\sqrt{2 + p}$ to both sides of the equation.

$\sqrt{11 - p} = - 1 + \sqrt{2 + p}$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{11 - p}}^{2} = {(- 1 + \sqrt{2 + p})}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 - p}$ as ${(11 - p)}^{\frac{1}{2}}$ .

${({(11 - p)}^{\frac{1}{2}})}^{2} = {(- 1 + \sqrt{2 + p})}^{2}$

Simplify the left side.

Simplify ${({(11 - p)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(11 - p)}^{\frac{1}{2}})}^{2}$ .

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

${(11 - p)}^{\frac{1}{2} \cdot 2} = {(- 1 + \sqrt{2 + p})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(11 - p)}^{\frac{1}{2} \cdot 2} = {(- 1 + \sqrt{2 + p})}^{2}$

Rewrite the expression.

${(11 - p)}^{1} = {(- 1 + \sqrt{2 + p})}^{2}$

Simplify.

$11 - p = {(- 1 + \sqrt{2 + p})}^{2}$

Simplify the right side.

Simplify ${(- 1 + \sqrt{2 + p})}^{2}$ .

Rewrite ${(- 1 + \sqrt{2 + p})}^{2}$ as $(- 1 + \sqrt{2 + p}) (- 1 + \sqrt{2 + p})$ .

$11 - p = (- 1 + \sqrt{2 + p}) (- 1 + \sqrt{2 + p})$

Expand $(- 1 + \sqrt{2 + p}) (- 1 + \sqrt{2 + p})$ using the FOIL Method.

Apply the distributive property.

$11 - p = - 1 (- 1 + \sqrt{2 + p}) + \sqrt{2 + p} (- 1 + \sqrt{2 + p})$

Apply the distributive property.

$11 - p = - 1 \cdot - 1 - 1 \sqrt{2 + p} + \sqrt{2 + p} (- 1 + \sqrt{2 + p})$

Apply the distributive property.

$11 - p = - 1 \cdot - 1 - 1 \sqrt{2 + p} + \sqrt{2 + p} \cdot - 1 + \sqrt{2 + p} \sqrt{2 + p}$

Simplify and combine like terms.

Simplify each term.

Multiply $- 1$ by $- 1$ .

$11 - p = 1 - 1 \sqrt{2 + p} + \sqrt{2 + p} \cdot - 1 + \sqrt{2 + p} \sqrt{2 + p}$

Rewrite $- 1 \sqrt{2 + p}$ as $- \sqrt{2 + p}$ .

$11 - p = 1 - \sqrt{2 + p} + \sqrt{2 + p} \cdot - 1 + \sqrt{2 + p} \sqrt{2 + p}$

Move $- 1$ to the left of $\sqrt{2 + p}$ .

$11 - p = 1 - \sqrt{2 + p} - 1 \cdot \sqrt{2 + p} + \sqrt{2 + p} \sqrt{2 + p}$

Rewrite $- 1 \sqrt{2 + p}$ as $- \sqrt{2 + p}$ .

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + \sqrt{2 + p} \sqrt{2 + p}$

Multiply $\sqrt{2 + p} \sqrt{2 + p}$ .

Raise $\sqrt{2 + p}$ to the power of $1$ .

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {\sqrt{2 + p}}^{1} \sqrt{2 + p}$

Raise $\sqrt{2 + p}$ to the power of $1$ .

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {\sqrt{2 + p}}^{1} {\sqrt{2 + p}}^{1}$

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

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {\sqrt{2 + p}}^{1 + 1}$

Add $1$ and $1$ .

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {\sqrt{2 + p}}^{2}$

Rewrite ${\sqrt{2 + p}}^{2}$ as $2 + p$ .

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

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {({(2 + p)}^{\frac{1}{2}})}^{2}$

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

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {(2 + p)}^{\frac{1}{2} \cdot 2}$

Combine $\frac{1}{2}$ and $2$ .

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {(2 + p)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {(2 + p)}^{\frac{2}{2}}$

Rewrite the expression.

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + {(2 + p)}^{1}$

Simplify.

$11 - p = 1 - \sqrt{2 + p} - \sqrt{2 + p} + 2 + p$

Add $1$ and $2$ .

$11 - p = 3 - \sqrt{2 + p} - \sqrt{2 + p} + p$

Subtract $\sqrt{2 + p}$ from $- \sqrt{2 + p}$ .

$11 - p = 3 - 2 \sqrt{2 + p} + p$

Solve for $- 2 \sqrt{2 + p}$ .

Rewrite the equation as $3 - 2 \sqrt{2 + p} + p = 11 - p$ .

$3 - 2 \sqrt{2 + p} + p = 11 - p$

Move all terms not containing $- 2 \sqrt{2 + p}$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$- 2 \sqrt{2 + p} + p = 11 - p - 3$

Subtract $p$ from both sides of the equation.

$- 2 \sqrt{2 + p} = 11 - p - 3 - p$

Subtract $3$ from $11$ .

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

Subtract $p$ from $- p$ .

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

To remove the radical on the left side of the equation, square both sides of the equation.

${(- 2 \sqrt{2 + p})}^{2} = {(- 2 p + 8)}^{2}$

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the product rule to $- 2 {(2 + p)}^{\frac{1}{2}}$ .

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

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

$4 {({(2 + p)}^{\frac{1}{2}})}^{2} = {(- 2 p + 8)}^{2}$

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

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

$4 {(2 + p)}^{\frac{1}{2} \cdot 2} = {(- 2 p + 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(2 + p)}^{\frac{1}{2} \cdot 2} = {(- 2 p + 8)}^{2}$

Rewrite the expression.

$4 {(2 + p)}^{1} = {(- 2 p + 8)}^{2}$

Simplify.

$4 (2 + p) = {(- 2 p + 8)}^{2}$

Apply the distributive property.

$4 \cdot 2 + 4 p = {(- 2 p + 8)}^{2}$

Multiply $4$ by $2$ .

$8 + 4 p = {(- 2 p + 8)}^{2}$

Simplify the right side.

Simplify ${(- 2 p + 8)}^{2}$ .

Rewrite ${(- 2 p + 8)}^{2}$ as $(- 2 p + 8) (- 2 p + 8)$ .

$8 + 4 p = (- 2 p + 8) (- 2 p + 8)$

Expand $(- 2 p + 8) (- 2 p + 8)$ using the FOIL Method.

Apply the distributive property.

$8 + 4 p = - 2 p (- 2 p + 8) + 8 (- 2 p + 8)$

Apply the distributive property.

$8 + 4 p = - 2 p (- 2 p) - 2 p \cdot 8 + 8 (- 2 p + 8)$

Apply the distributive property.

$8 + 4 p = - 2 p (- 2 p) - 2 p \cdot 8 + 8 (- 2 p) + 8 \cdot 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 + 4 p = - 2 \cdot - 2 p \cdot p - 2 p \cdot 8 + 8 (- 2 p) + 8 \cdot 8$

Multiply $p$ by $p$ by adding the exponents.

Move $p$ .

$8 + 4 p = - 2 \cdot - 2 (p \cdot p) - 2 p \cdot 8 + 8 (- 2 p) + 8 \cdot 8$

Multiply $p$ by $p$ .

$8 + 4 p = - 2 \cdot - 2 p^{2} - 2 p \cdot 8 + 8 (- 2 p) + 8 \cdot 8$

Multiply $- 2$ by $- 2$ .

$8 + 4 p = 4 p^{2} - 2 p \cdot 8 + 8 (- 2 p) + 8 \cdot 8$

Multiply $8$ by $- 2$ .

$8 + 4 p = 4 p^{2} - 16 p + 8 (- 2 p) + 8 \cdot 8$

Multiply $- 2$ by $8$ .

$8 + 4 p = 4 p^{2} - 16 p - 16 p + 8 \cdot 8$

Multiply $8$ by $8$ .

$8 + 4 p = 4 p^{2} - 16 p - 16 p + 64$

Subtract $16 p$ from $- 16 p$ .

$8 + 4 p = 4 p^{2} - 32 p + 64$

Solve for $p$ .

Since $p$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$4 p^{2} - 32 p + 64 = 8 + 4 p$

Move all terms containing $p$ to the left side of the equation.

Subtract $4 p$ from both sides of the equation.

$4 p^{2} - 32 p + 64 - 4 p = 8$

Subtract $4 p$ from $- 32 p$ .

$4 p^{2} - 36 p + 64 = 8$

Subtract $8$ from both sides of the equation.

$4 p^{2} - 36 p + 64 - 8 = 0$

Subtract $8$ from $64$ .

$4 p^{2} - 36 p + 56 = 0$

Factor the left side of the equation.

Factor $4$ out of $4 p^{2} - 36 p + 56$ .

Factor $4$ out of $4 p^{2}$ .

$4 (p^{2}) - 36 p + 56 = 0$

Factor $4$ out of $- 36 p$ .

$4 (p^{2}) + 4 (- 9 p) + 56 = 0$

Factor $4$ out of $56$ .

$4 p^{2} + 4 (- 9 p) + 4 \cdot 14 = 0$

Factor $4$ out of $4 p^{2} + 4 (- 9 p)$ .

$4 (p^{2} - 9 p) + 4 \cdot 14 = 0$

Factor $4$ out of $4 (p^{2} - 9 p) + 4 \cdot 14$ .

$4 (p^{2} - 9 p + 14) = 0$

Factor.

Factor $p^{2} - 9 p + 14$ 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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Write the factored form using these integers.

$4 ((p - 7) (p - 2)) = 0$

Remove unnecessary parentheses.

$4 (p - 7) (p - 2) = 0$

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

$p - 7 = 0$

$p - 2 = 0$

Set $p - 7$ equal to $0$ and solve for $p$ .

Set $p - 7$ equal to $0$ .

$p - 7 = 0$

Add $7$ to both sides of the equation.

$p = 7$

Set $p - 2$ equal to $0$ and solve for $p$ .

Set $p - 2$ equal to $0$ .

$p - 2 = 0$

Add $2$ to both sides of the equation.

$p = 2$

The final solution is all the values that make $4 (p - 7) (p - 2) = 0$ true.

$p = 7, 2$

Exclude the solutions that do not make $\sqrt{11 - p} - \sqrt{2 + p} = - 1$ true.

$p = 7$

Solve for p square root of 11-p- square root of 2+p=-1

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