Solve by Factoring 3+ square root of z-6=- square root of z+9

$3 + \sqrt{z - 6} = - \sqrt{z + 9}$

Step 1

Add $\sqrt{z + 9}$ to both sides of the equation.

$3 + \sqrt{z - 6} + \sqrt{z + 9} = 0$

Step 2

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

Subtract $3$ from both sides of the equation.

$\sqrt{z - 6} + \sqrt{z + 9} = - 3$

Subtract $\sqrt{z + 9}$ from both sides of the equation.

$\sqrt{z - 6} = - 3 - \sqrt{z + 9}$

Step 3

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

${\sqrt{z - 6}}^{2} = {(- 3 - \sqrt{z + 9})}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 6}$ as ${(z - 6)}^{\frac{1}{2}}$ .

${({(z - 6)}^{\frac{1}{2}})}^{2} = {(- 3 - \sqrt{z + 9})}^{2}$

Simplify the left side.

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

Multiply the exponents in ${({(z - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(z - 6)}^{\frac{1}{2} \cdot 2} = {(- 3 - \sqrt{z + 9})}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(z - 6)}^{\frac{1}{2} \cdot 2} = {(- 3 - \sqrt{z + 9})}^{2}$

Rewrite the expression.

${(z - 6)}^{1} = {(- 3 - \sqrt{z + 9})}^{2}$

Simplify.

$z - 6 = {(- 3 - \sqrt{z + 9})}^{2}$

Simplify the right side.

Simplify ${(- 3 - \sqrt{z + 9})}^{2}$ .

Rewrite ${(- 3 - \sqrt{z + 9})}^{2}$ as $(- 3 - \sqrt{z + 9}) (- 3 - \sqrt{z + 9})$ .

$z - 6 = (- 3 - \sqrt{z + 9}) (- 3 - \sqrt{z + 9})$

Expand $(- 3 - \sqrt{z + 9}) (- 3 - \sqrt{z + 9})$ using the FOIL Method.

Apply the distributive property.

$z - 6 = - 3 (- 3 - \sqrt{z + 9}) - \sqrt{z + 9} (- 3 - \sqrt{z + 9})$

Apply the distributive property.

$z - 6 = - 3 \cdot - 3 - 3 (- \sqrt{z + 9}) - \sqrt{z + 9} (- 3 - \sqrt{z + 9})$

Apply the distributive property.

$z - 6 = - 3 \cdot - 3 - 3 (- \sqrt{z + 9}) - \sqrt{z + 9} \cdot - 3 - \sqrt{z + 9} (- \sqrt{z + 9})$

Simplify and combine like terms.

Simplify each term.

Multiply $- 3$ by $- 3$ .

$z - 6 = 9 - 3 (- \sqrt{z + 9}) - \sqrt{z + 9} \cdot - 3 - \sqrt{z + 9} (- \sqrt{z + 9})$

Multiply $- 1$ by $- 3$ .

$z - 6 = 9 + 3 \sqrt{z + 9} - \sqrt{z + 9} \cdot - 3 - \sqrt{z + 9} (- \sqrt{z + 9})$

Multiply $- 3$ by $- 1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} - \sqrt{z + 9} (- \sqrt{z + 9})$

Multiply $- \sqrt{z + 9} (- \sqrt{z + 9})$ .

Multiply $- 1$ by $- 1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + 1 \sqrt{z + 9} \sqrt{z + 9}$

Multiply $\sqrt{z + 9}$ by $1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + \sqrt{z + 9} \sqrt{z + 9}$

Raise $\sqrt{z + 9}$ to the power of $1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {\sqrt{z + 9}}^{1} \sqrt{z + 9}$

Raise $\sqrt{z + 9}$ to the power of $1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {\sqrt{z + 9}}^{1} {\sqrt{z + 9}}^{1}$

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

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {\sqrt{z + 9}}^{1 + 1}$

Add $1$ and $1$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {\sqrt{z + 9}}^{2}$

Rewrite ${\sqrt{z + 9}}^{2}$ as $z + 9$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z + 9}$ as ${(z + 9)}^{\frac{1}{2}}$ .

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {({(z + 9)}^{\frac{1}{2}})}^{2}$

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

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {(z + 9)}^{\frac{1}{2} \cdot 2}$

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

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {(z + 9)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {(z + 9)}^{\frac{2}{2}}$

Rewrite the expression.

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + {(z + 9)}^{1}$

Simplify.

$z - 6 = 9 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + z + 9$

Add $9$ and $9$ .

$z - 6 = 18 + 3 \sqrt{z + 9} + 3 \sqrt{z + 9} + z$

Add $3 \sqrt{z + 9}$ and $3 \sqrt{z + 9}$ .

$z - 6 = 18 + 6 \sqrt{z + 9} + z$

Step 5

Solve for $6 \sqrt{z + 9}$ .

Rewrite the equation as $18 + 6 \sqrt{z + 9} + z = z - 6$ .

$18 + 6 \sqrt{z + 9} + z = z - 6$

Move all terms not containing $6 \sqrt{z + 9}$ to the right side of the equation.

Subtract $18$ from both sides of the equation.

$6 \sqrt{z + 9} + z = z - 6 - 18$

Subtract $z$ from both sides of the equation.

$6 \sqrt{z + 9} = z - 6 - 18 - z$

Combine the opposite terms in $z - 6 - 18 - z$ .

Subtract $z$ from $z$ .

$6 \sqrt{z + 9} = 0 - 6 - 18$

Subtract $6$ from $0$ .

$6 \sqrt{z + 9} = - 6 - 18$

Subtract $18$ from $- 6$ .

$6 \sqrt{z + 9} = - 24$

Step 6

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

${(6 \sqrt{z + 9})}^{2} = {(- 24)}^{2}$

Step 7

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z + 9}$ as ${(z + 9)}^{\frac{1}{2}}$ .

${(6 {(z + 9)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Simplify the left side.

Simplify ${(6 {(z + 9)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $6 {(z + 9)}^{\frac{1}{2}}$ .

$6^{2} {({(z + 9)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

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

$36 {({(z + 9)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Multiply the exponents in ${({(z + 9)}^{\frac{1}{2}})}^{2}$ .

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

$36 {(z + 9)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(z + 9)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Rewrite the expression.

$36 {(z + 9)}^{1} = {(- 24)}^{2}$

Simplify.

$36 (z + 9) = {(- 24)}^{2}$

Apply the distributive property.

$36 z + 36 \cdot 9 = {(- 24)}^{2}$

Multiply $36$ by $9$ .

$36 z + 324 = {(- 24)}^{2}$

Simplify the right side.

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

$36 z + 324 = 576$

Step 8

Solve for $z$ .

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

Subtract $324$ from both sides of the equation.

$36 z = 576 - 324$

Subtract $324$ from $576$ .

$36 z = 252$

Divide each term in $36 z = 252$ by $36$ and simplify.

Divide each term in $36 z = 252$ by $36$ .

$\frac{36 z}{36} = \frac{252}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 z}{36} = \frac{252}{36}$

Divide $z$ by $1$ .

$z = \frac{252}{36}$

Simplify the right side.

Divide $252$ by $36$ .

$z = 7$

Step 9

Exclude the solutions that do not make $3 + \sqrt{z - 6} + \sqrt{z + 9} = 0$ true.

$No solution$

Solve by Factoring 3+ square root of z-6=- square root of z+9

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