Solve for p square root of 8-p=2+ square root of 2p+3

$\sqrt{8 - p} = 2 + \sqrt{2 p + 3}$

Step 1

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

${\sqrt{8 - p}}^{2} = {(2 + \sqrt{2 p + 3})}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - p}$ as ${(8 - p)}^{\frac{1}{2}}$ .

${({(8 - p)}^{\frac{1}{2}})}^{2} = {(2 + \sqrt{2 p + 3})}^{2}$

Simplify the left side.

Simplify ${({(8 - p)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8 - p)}^{\frac{1}{2}})}^{2}$ .

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

${(8 - p)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{2 p + 3})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 - p)}^{\frac{1}{2} \cdot 2} = {(2 + \sqrt{2 p + 3})}^{2}$

Rewrite the expression.

${(8 - p)}^{1} = {(2 + \sqrt{2 p + 3})}^{2}$

Simplify.

$8 - p = {(2 + \sqrt{2 p + 3})}^{2}$

Simplify the right side.

Simplify ${(2 + \sqrt{2 p + 3})}^{2}$ .

Rewrite ${(2 + \sqrt{2 p + 3})}^{2}$ as $(2 + \sqrt{2 p + 3}) (2 + \sqrt{2 p + 3})$ .

$8 - p = (2 + \sqrt{2 p + 3}) (2 + \sqrt{2 p + 3})$

Expand $(2 + \sqrt{2 p + 3}) (2 + \sqrt{2 p + 3})$ using the FOIL Method.

Apply the distributive property.

$8 - p = 2 (2 + \sqrt{2 p + 3}) + \sqrt{2 p + 3} (2 + \sqrt{2 p + 3})$

Apply the distributive property.

$8 - p = 2 \cdot 2 + 2 \sqrt{2 p + 3} + \sqrt{2 p + 3} (2 + \sqrt{2 p + 3})$

Apply the distributive property.

$8 - p = 2 \cdot 2 + 2 \sqrt{2 p + 3} + \sqrt{2 p + 3} \cdot 2 + \sqrt{2 p + 3} \sqrt{2 p + 3}$

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + \sqrt{2 p + 3} \cdot 2 + \sqrt{2 p + 3} \sqrt{2 p + 3}$

Move $2$ to the left of $\sqrt{2 p + 3}$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \cdot \sqrt{2 p + 3} + \sqrt{2 p + 3} \sqrt{2 p + 3}$

Multiply $\sqrt{2 p + 3} \sqrt{2 p + 3}$ .

Raise $\sqrt{2 p + 3}$ to the power of $1$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {\sqrt{2 p + 3}}^{1} \sqrt{2 p + 3}$

Raise $\sqrt{2 p + 3}$ to the power of $1$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {\sqrt{2 p + 3}}^{1} {\sqrt{2 p + 3}}^{1}$

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

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {\sqrt{2 p + 3}}^{1 + 1}$

Add $1$ and $1$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {\sqrt{2 p + 3}}^{2}$

Rewrite ${\sqrt{2 p + 3}}^{2}$ as $2 p + 3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 p + 3}$ as ${(2 p + 3)}^{\frac{1}{2}}$ .

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {({(2 p + 3)}^{\frac{1}{2}})}^{2}$

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

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {(2 p + 3)}^{\frac{1}{2} \cdot 2}$

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

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {(2 p + 3)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {(2 p + 3)}^{\frac{2}{2}}$

Rewrite the expression.

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + {(2 p + 3)}^{1}$

Simplify.

$8 - p = 4 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + 2 p + 3$

Add $4$ and $3$ .

$8 - p = 7 + 2 \sqrt{2 p + 3} + 2 \sqrt{2 p + 3} + 2 p$

Add $2 \sqrt{2 p + 3}$ and $2 \sqrt{2 p + 3}$ .

$8 - p = 7 + 4 \sqrt{2 p + 3} + 2 p$

Step 3

Solve for $4 \sqrt{2 p + 3}$ .

Rewrite the equation as $7 + 4 \sqrt{2 p + 3} + 2 p = 8 - p$ .

$7 + 4 \sqrt{2 p + 3} + 2 p = 8 - p$

Move all terms not containing $4 \sqrt{2 p + 3}$ to the right side of the equation.

Subtract $7$ from both sides of the equation.

$4 \sqrt{2 p + 3} + 2 p = 8 - p - 7$

Subtract $2 p$ from both sides of the equation.

$4 \sqrt{2 p + 3} = 8 - p - 7 - 2 p$

Subtract $7$ from $8$ .

$4 \sqrt{2 p + 3} = - p + 1 - 2 p$

Subtract $2 p$ from $- p$ .

$4 \sqrt{2 p + 3} = - 3 p + 1$

Step 4

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

${(4 \sqrt{2 p + 3})}^{2} = {(- 3 p + 1)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 p + 3}$ as ${(2 p + 3)}^{\frac{1}{2}}$ .

${(4 {(2 p + 3)}^{\frac{1}{2}})}^{2} = {(- 3 p + 1)}^{2}$

Simplify the left side.

Simplify ${(4 {(2 p + 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $4 {(2 p + 3)}^{\frac{1}{2}}$ .

$4^{2} {({(2 p + 3)}^{\frac{1}{2}})}^{2} = {(- 3 p + 1)}^{2}$

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

$16 {({(2 p + 3)}^{\frac{1}{2}})}^{2} = {(- 3 p + 1)}^{2}$

Multiply the exponents in ${({(2 p + 3)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(2 p + 3)}^{\frac{1}{2} \cdot 2} = {(- 3 p + 1)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(2 p + 3)}^{\frac{1}{2} \cdot 2} = {(- 3 p + 1)}^{2}$

Rewrite the expression.

$16 {(2 p + 3)}^{1} = {(- 3 p + 1)}^{2}$

Simplify.

$16 (2 p + 3) = {(- 3 p + 1)}^{2}$

Apply the distributive property.

$16 (2 p) + 16 \cdot 3 = {(- 3 p + 1)}^{2}$

Multiply.

Multiply $2$ by $16$ .

$32 p + 16 \cdot 3 = {(- 3 p + 1)}^{2}$

Multiply $16$ by $3$ .

$32 p + 48 = {(- 3 p + 1)}^{2}$

Simplify the right side.

Simplify ${(- 3 p + 1)}^{2}$ .

Rewrite ${(- 3 p + 1)}^{2}$ as $(- 3 p + 1) (- 3 p + 1)$ .

$32 p + 48 = (- 3 p + 1) (- 3 p + 1)$

Expand $(- 3 p + 1) (- 3 p + 1)$ using the FOIL Method.

Apply the distributive property.

$32 p + 48 = - 3 p (- 3 p + 1) + 1 (- 3 p + 1)$

Apply the distributive property.

$32 p + 48 = - 3 p (- 3 p) - 3 p \cdot 1 + 1 (- 3 p + 1)$

Apply the distributive property.

$32 p + 48 = - 3 p (- 3 p) - 3 p \cdot 1 + 1 (- 3 p) + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$32 p + 48 = - 3 \cdot - 3 p \cdot p - 3 p \cdot 1 + 1 (- 3 p) + 1 \cdot 1$

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

Move $p$ .

$32 p + 48 = - 3 \cdot - 3 (p \cdot p) - 3 p \cdot 1 + 1 (- 3 p) + 1 \cdot 1$

Multiply $p$ by $p$ .

$32 p + 48 = - 3 \cdot - 3 p^{2} - 3 p \cdot 1 + 1 (- 3 p) + 1 \cdot 1$

Multiply $- 3$ by $- 3$ .

$32 p + 48 = 9 p^{2} - 3 p \cdot 1 + 1 (- 3 p) + 1 \cdot 1$

Multiply $- 3$ by $1$ .

$32 p + 48 = 9 p^{2} - 3 p + 1 (- 3 p) + 1 \cdot 1$

Multiply $- 3 p$ by $1$ .

$32 p + 48 = 9 p^{2} - 3 p - 3 p + 1 \cdot 1$

Multiply $1$ by $1$ .

$32 p + 48 = 9 p^{2} - 3 p - 3 p + 1$

Subtract $3 p$ from $- 3 p$ .

$32 p + 48 = 9 p^{2} - 6 p + 1$

Step 6

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.

$9 p^{2} - 6 p + 1 = 32 p + 48$

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

Subtract $32 p$ from both sides of the equation.

$9 p^{2} - 6 p + 1 - 32 p = 48$

Subtract $32 p$ from $- 6 p$ .

$9 p^{2} - 38 p + 1 = 48$

Subtract $48$ from both sides of the equation.

$9 p^{2} - 38 p + 1 - 48 = 0$

Subtract $48$ from $1$ .

$9 p^{2} - 38 p - 47 = 0$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 47 = - 423$ and whose sum is $b = - 38$ .

Factor $- 38$ out of $- 38 p$ .

$9 p^{2} - 38 p - 47 = 0$

Rewrite $- 38$ as $9$ plus $- 47$

$9 p^{2} + (9 - 47) p - 47 = 0$

Apply the distributive property.

$9 p^{2} + 9 p - 47 p - 47 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(9 p^{2} + 9 p) - 47 p - 47 = 0$

Factor out the greatest common factor (GCF) from each group.

$9 p (p + 1) - 47 (p + 1) = 0$

Factor the polynomial by factoring out the greatest common factor, $p + 1$ .

$(p + 1) (9 p - 47) = 0$

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

$p + 1 = 0$

$9 p - 47 = 0$

Set $p + 1$ equal to $0$ and solve for $p$ .

Set $p + 1$ equal to $0$ .

$p + 1 = 0$

Subtract $1$ from both sides of the equation.

$p = - 1$

Set $9 p - 47$ equal to $0$ and solve for $p$ .

Set $9 p - 47$ equal to $0$ .

$9 p - 47 = 0$

Solve $9 p - 47 = 0$ for $p$ .

Add $47$ to both sides of the equation.

$9 p = 47$

Divide each term in $9 p = 47$ by $9$ and simplify.

Divide each term in $9 p = 47$ by $9$ .

$\frac{9 p}{9} = \frac{47}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 p}{9} = \frac{47}{9}$

Divide $p$ by $1$ .

$p = \frac{47}{9}$

The final solution is all the values that make $(p + 1) (9 p - 47) = 0$ true.

$p = - 1, \frac{47}{9}$

Step 7

Exclude the solutions that do not make $\sqrt{8 - p} = 2 + \sqrt{2 p + 3}$ true.

$p = - 1$

Solve for p square root of 8-p=2+ square root of 2p+3

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