Solve by Factoring 3+ square root of z-4=- square root of z+11

$3 + \sqrt{z - 4} = - \sqrt{z + 11}$

Step 1

Add $\sqrt{z + 11}$ to both sides of the equation.

$3 + \sqrt{z - 4} + \sqrt{z + 11} = 0$

Step 2

Move all terms not containing $\sqrt{z - 4}$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$\sqrt{z - 4} + \sqrt{z + 11} = - 3$

Subtract $\sqrt{z + 11}$ from both sides of the equation.

$\sqrt{z - 4} = - 3 - \sqrt{z + 11}$

Step 3

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

${\sqrt{z - 4}}^{2} = {(- 3 - \sqrt{z + 11})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 4}$ as ${(z - 4)}^{\frac{1}{2}}$ .

${({(z - 4)}^{\frac{1}{2}})}^{2} = {(- 3 - \sqrt{z + 11})}^{2}$

Simplify the left side.

Simplify ${({(z - 4)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(z - 4)}^{\frac{1}{2}})}^{2}$ .

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

${(z - 4)}^{\frac{1}{2} \cdot 2} = {(- 3 - \sqrt{z + 11})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(z - 4)}^{\frac{1}{2} \cdot 2} = {(- 3 - \sqrt{z + 11})}^{2}$

Rewrite the expression.

${(z - 4)}^{1} = {(- 3 - \sqrt{z + 11})}^{2}$

Simplify.

$z - 4 = {(- 3 - \sqrt{z + 11})}^{2}$

Simplify the right side.

Simplify ${(- 3 - \sqrt{z + 11})}^{2}$ .

Rewrite ${(- 3 - \sqrt{z + 11})}^{2}$ as $(- 3 - \sqrt{z + 11}) (- 3 - \sqrt{z + 11})$ .

$z - 4 = (- 3 - \sqrt{z + 11}) (- 3 - \sqrt{z + 11})$

Expand $(- 3 - \sqrt{z + 11}) (- 3 - \sqrt{z + 11})$ using the FOIL Method.

Apply the distributive property.

$z - 4 = - 3 (- 3 - \sqrt{z + 11}) - \sqrt{z + 11} (- 3 - \sqrt{z + 11})$

Apply the distributive property.

$z - 4 = - 3 \cdot - 3 - 3 (- \sqrt{z + 11}) - \sqrt{z + 11} (- 3 - \sqrt{z + 11})$

Apply the distributive property.

$z - 4 = - 3 \cdot - 3 - 3 (- \sqrt{z + 11}) - \sqrt{z + 11} \cdot - 3 - \sqrt{z + 11} (- \sqrt{z + 11})$

Simplify and combine like terms.

Simplify each term.

Multiply $- 3$ by $- 3$ .

$z - 4 = 9 - 3 (- \sqrt{z + 11}) - \sqrt{z + 11} \cdot - 3 - \sqrt{z + 11} (- \sqrt{z + 11})$

Multiply $- 1$ by $- 3$ .

$z - 4 = 9 + 3 \sqrt{z + 11} - \sqrt{z + 11} \cdot - 3 - \sqrt{z + 11} (- \sqrt{z + 11})$

Multiply $- 3$ by $- 1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} - \sqrt{z + 11} (- \sqrt{z + 11})$

Multiply $- \sqrt{z + 11} (- \sqrt{z + 11})$ .

Multiply $- 1$ by $- 1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + 1 \sqrt{z + 11} \sqrt{z + 11}$

Multiply $\sqrt{z + 11}$ by $1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + \sqrt{z + 11} \sqrt{z + 11}$

Raise $\sqrt{z + 11}$ to the power of $1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {\sqrt{z + 11}}^{1} \sqrt{z + 11}$

Raise $\sqrt{z + 11}$ to the power of $1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {\sqrt{z + 11}}^{1} {\sqrt{z + 11}}^{1}$

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

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {\sqrt{z + 11}}^{1 + 1}$

Add $1$ and $1$ .

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {\sqrt{z + 11}}^{2}$

Rewrite ${\sqrt{z + 11}}^{2}$ as $z + 11$ .

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

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {({(z + 11)}^{\frac{1}{2}})}^{2}$

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

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {(z + 11)}^{\frac{1}{2} \cdot 2}$

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

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {(z + 11)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {(z + 11)}^{\frac{2}{2}}$

Rewrite the expression.

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + {(z + 11)}^{1}$

Simplify.

$z - 4 = 9 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + z + 11$

Add $9$ and $11$ .

$z - 4 = 20 + 3 \sqrt{z + 11} + 3 \sqrt{z + 11} + z$

Add $3 \sqrt{z + 11}$ and $3 \sqrt{z + 11}$ .

$z - 4 = 20 + 6 \sqrt{z + 11} + z$

Step 5

Solve for $6 \sqrt{z + 11}$ .

Rewrite the equation as $20 + 6 \sqrt{z + 11} + z = z - 4$ .

$20 + 6 \sqrt{z + 11} + z = z - 4$

Move all terms not containing $6 \sqrt{z + 11}$ to the right side of the equation.

Subtract $20$ from both sides of the equation.

$6 \sqrt{z + 11} + z = z - 4 - 20$

Subtract $z$ from both sides of the equation.

$6 \sqrt{z + 11} = z - 4 - 20 - z$

Combine the opposite terms in $z - 4 - 20 - z$ .

Subtract $z$ from $z$ .

$6 \sqrt{z + 11} = 0 - 4 - 20$

Subtract $4$ from $0$ .

$6 \sqrt{z + 11} = - 4 - 20$

Subtract $20$ from $- 4$ .

$6 \sqrt{z + 11} = - 24$

Step 6

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

${(6 \sqrt{z + 11})}^{2} = {(- 24)}^{2}$

Step 7

Simplify each side of the equation.

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

${(6 {(z + 11)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Simplify the left side.

Simplify ${(6 {(z + 11)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $6 {(z + 11)}^{\frac{1}{2}}$ .

$6^{2} {({(z + 11)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Raise $6$ to the power of $2$ .

$36 {({(z + 11)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

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

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

$36 {(z + 11)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(z + 11)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Rewrite the expression.

$36 {(z + 11)}^{1} = {(- 24)}^{2}$

Simplify.

$36 (z + 11) = {(- 24)}^{2}$

Apply the distributive property.

$36 z + 36 \cdot 11 = {(- 24)}^{2}$

Multiply $36$ by $11$ .

$36 z + 396 = {(- 24)}^{2}$

Simplify the right side.

Raise $- 24$ to the power of $2$ .

$36 z + 396 = 576$

Step 8

Solve for $z$ .

Move all terms not containing $z$ to the right side of the equation.

Subtract $396$ from both sides of the equation.

$36 z = 576 - 396$

Subtract $396$ from $576$ .

$36 z = 180$

Divide each term in $36 z = 180$ by $36$ and simplify.

Divide each term in $36 z = 180$ by $36$ .

$\frac{36 z}{36} = \frac{180}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 z}{36} = \frac{180}{36}$

Divide $z$ by $1$ .

$z = \frac{180}{36}$

Simplify the right side.

Divide $180$ by $36$ .

$z = 5$

Step 9

Exclude the solutions that do not make $3 + \sqrt{z - 4} + \sqrt{z + 11} = 0$ true.

$No solution$

Solve by Factoring 3+ square root of z-4=- square root of z+11

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