Solve for z 3+ square root of z-12=- square root of z+3

$3 + \sqrt{z - 12} = - \sqrt{z + 3}$

Step 1

Subtract $3$ from both sides of the equation.

$\sqrt{z - 12} = - \sqrt{z + 3} - 3$

Step 2

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

${\sqrt{z - 12}}^{2} = {(- \sqrt{z + 3} - 3)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z - 12}$ as ${(z - 12)}^{\frac{1}{2}}$ .

${({(z - 12)}^{\frac{1}{2}})}^{2} = {(- \sqrt{z + 3} - 3)}^{2}$

Simplify the left side.

Simplify ${({(z - 12)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(z - 12)}^{\frac{1}{2}})}^{2}$ .

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

${(z - 12)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{z + 3} - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(z - 12)}^{\frac{1}{2} \cdot 2} = {(- \sqrt{z + 3} - 3)}^{2}$

Rewrite the expression.

${(z - 12)}^{1} = {(- \sqrt{z + 3} - 3)}^{2}$

Simplify.

$z - 12 = {(- \sqrt{z + 3} - 3)}^{2}$

Simplify the right side.

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

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

$z - 12 = (- \sqrt{z + 3} - 3) (- \sqrt{z + 3} - 3)$

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

Apply the distributive property.

$z - 12 = - \sqrt{z + 3} (- \sqrt{z + 3} - 3) - 3 (- \sqrt{z + 3} - 3)$

Apply the distributive property.

$z - 12 = - \sqrt{z + 3} (- \sqrt{z + 3}) - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3} - 3)$

Apply the distributive property.

$z - 12 = - \sqrt{z + 3} (- \sqrt{z + 3}) - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $- \sqrt{z + 3} (- \sqrt{z + 3})$ .

Multiply $- 1$ by $- 1$ .

$z - 12 = 1 \sqrt{z + 3} \sqrt{z + 3} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Multiply $\sqrt{z + 3}$ by $1$ .

$z - 12 = \sqrt{z + 3} \sqrt{z + 3} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

$z - 12 = {\sqrt{z + 3}}^{1} \sqrt{z + 3} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

$z - 12 = {\sqrt{z + 3}}^{1} {\sqrt{z + 3}}^{1} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

$z - 12 = {\sqrt{z + 3}}^{1 + 1} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Add $1$ and $1$ .

$z - 12 = {\sqrt{z + 3}}^{2} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

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

$z - 12 = {({(z + 3)}^{\frac{1}{2}})}^{2} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

$z - 12 = {(z + 3)}^{\frac{1}{2} \cdot 2} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

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

$z - 12 = {(z + 3)}^{\frac{2}{2}} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$z - 12 = {(z + 3)}^{\frac{2}{2}} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Rewrite the expression.

$z - 12 = {(z + 3)}^{1} - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Simplify.

$z - 12 = z + 3 - \sqrt{z + 3} \cdot - 3 - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Multiply $- 3$ by $- 1$ .

$z - 12 = z + 3 + 3 \sqrt{z + 3} - 3 (- \sqrt{z + 3}) - 3 \cdot - 3$

Multiply $- 1$ by $- 3$ .

$z - 12 = z + 3 + 3 \sqrt{z + 3} + 3 \sqrt{z + 3} - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$z - 12 = z + 3 + 3 \sqrt{z + 3} + 3 \sqrt{z + 3} + 9$

Add $3$ and $9$ .

$z - 12 = z + 12 + 3 \sqrt{z + 3} + 3 \sqrt{z + 3}$

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

$z - 12 = z + 12 + 6 \sqrt{z + 3}$

Step 4

Solve for $6 \sqrt{z + 3}$ .

Rewrite the equation as $z + 12 + 6 \sqrt{z + 3} = z - 12$ .

$z + 12 + 6 \sqrt{z + 3} = z - 12$

Move all terms not containing $6 \sqrt{z + 3}$ to the right side of the equation.

Subtract $z$ from both sides of the equation.

$12 + 6 \sqrt{z + 3} = z - 12 - z$

Subtract $12$ from both sides of the equation.

$6 \sqrt{z + 3} = z - 12 - z - 12$

Combine the opposite terms in $z - 12 - z - 12$ .

Subtract $z$ from $z$ .

$6 \sqrt{z + 3} = 0 - 12 - 12$

Subtract $12$ from $0$ .

$6 \sqrt{z + 3} = - 12 - 12$

Subtract $12$ from $- 12$ .

$6 \sqrt{z + 3} = - 24$

Step 5

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

${(6 \sqrt{z + 3})}^{2} = {(- 24)}^{2}$

Step 6

Simplify each side of the equation.

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

${(6 {(z + 3)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

Simplify the left side.

Simplify ${(6 {(z + 3)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $6 {(z + 3)}^{\frac{1}{2}}$ .

$6^{2} {({(z + 3)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

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

$36 {({(z + 3)}^{\frac{1}{2}})}^{2} = {(- 24)}^{2}$

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

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

$36 {(z + 3)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(z + 3)}^{\frac{1}{2} \cdot 2} = {(- 24)}^{2}$

Rewrite the expression.

$36 {(z + 3)}^{1} = {(- 24)}^{2}$

Simplify.

$36 (z + 3) = {(- 24)}^{2}$

Apply the distributive property.

$36 z + 36 \cdot 3 = {(- 24)}^{2}$

Multiply $36$ by $3$ .

$36 z + 108 = {(- 24)}^{2}$

Simplify the right side.

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

$36 z + 108 = 576$

Step 7

Solve for $z$ .

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

Subtract $108$ from both sides of the equation.

$36 z = 576 - 108$

Subtract $108$ from $576$ .

$36 z = 468$

Divide each term in $36 z = 468$ by $36$ and simplify.

Divide each term in $36 z = 468$ by $36$ .

$\frac{36 z}{36} = \frac{468}{36}$

Simplify the left side.

Cancel the common factor of $36$ .

Cancel the common factor.

$\frac{36 z}{36} = \frac{468}{36}$

Divide $z$ by $1$ .

$z = \frac{468}{36}$

Simplify the right side.

Divide $468$ by $36$ .

$z = 13$

Step 8

Exclude the solutions that do not make $3 + \sqrt{z - 12} = - \sqrt{z + 3}$ true.

$No solution$

Solve for z 3+ square root of z-12=- square root of z+3

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