Solve for z square root of 15-z+6 = square root of 4-9

$\sqrt{15 - z} + 6 = \sqrt{4 - 9}$

Simplify $\sqrt{4 - 9}$ .

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Subtract $9$ from $4$ .

$\sqrt{15 - z} + 6 = \sqrt{- 5}$

Rewrite $- 5$ as $- 1 (5)$ .

$\sqrt{15 - z} + 6 = \sqrt{- 1 (5)}$

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

$\sqrt{15 - z} + 6 = \sqrt{- 1} \cdot \sqrt{5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$\sqrt{15 - z} + 6 = i \sqrt{5}$

Move all terms not containing $\sqrt{15 - z}$ to the right side of the equation.

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Subtract $6$ from both sides of the equation.

$\sqrt{15 - z} = i \sqrt{5} - 6$

Reorder $i \sqrt{5}$ and $- 6$ .

$\sqrt{15 - z} = - 6 + i \sqrt{5}$

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

${\sqrt{15 - z}}^{2} = {(- 6 + i \sqrt{5})}^{2}$

Simplify each side of the equation.

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Multiply the exponents in ${({(15 - z)}^{\frac{1}{2}})}^{2}$ .

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

${(15 - z)}^{\frac{1}{2} \cdot 2} = {(- 6 + i \sqrt{5})}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(15 - z)}^{\frac{1}{2} \cdot 2} = {(- 6 + i \sqrt{5})}^{2}$

Rewrite the expression.

${(15 - z)}^{1} = {(- 6 + i \sqrt{5})}^{2}$

Simplify.

$15 - z = {(- 6 + i \sqrt{5})}^{2}$

Solve for $z$ .

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Simplify ${(- 6 + i \sqrt{5})}^{2}$ .

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Rewrite ${(- 6 + i \sqrt{5})}^{2}$ as $(- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$ .

$15 - z = (- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$

Expand $(- 6 + i \sqrt{5}) (- 6 + i \sqrt{5})$ using the FOIL Method.

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Apply the distributive property.

$15 - z = - 6 (- 6 + i \sqrt{5}) + i \sqrt{5} (- 6 + i \sqrt{5})$

Apply the distributive property.

$15 - z = - 6 \cdot - 6 - 6 (i \sqrt{5}) + i \sqrt{5} (- 6 + i \sqrt{5})$

Apply the distributive property.

$15 - z = - 6 \cdot - 6 - 6 (i \sqrt{5}) + i \sqrt{5} \cdot - 6 + i \sqrt{5} (i \sqrt{5})$

Simplify and combine like terms.

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Simplify each term.

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Multiply $- 6$ by $- 6$ .

$15 - z = 36 - 6 (i \sqrt{5}) + i \sqrt{5} \cdot - 6 + i \sqrt{5} (i \sqrt{5})$

Move $- 6$ to the left of $i \sqrt{5}$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 \cdot (i \sqrt{5}) + i \sqrt{5} (i \sqrt{5})$

Multiply $i \sqrt{5} (i \sqrt{5})$ .

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Raise $i$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1} i \sqrt{5} \sqrt{5}$

Raise $i$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1} i^{1} \sqrt{5} \sqrt{5}$

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{1 + 1} \sqrt{5} \sqrt{5}$

Add $1$ and $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} \sqrt{5} \sqrt{5}$

Raise $\sqrt{5}$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} \sqrt{5})$

Raise $\sqrt{5}$ to the power of $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} ({\sqrt{5}}^{1} {\sqrt{5}}^{1})$

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} {\sqrt{5}}^{1 + 1}$

Add $1$ and $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} + i^{2} {\sqrt{5}}^{2}$

Rewrite $i^{2}$ as $- 1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 {\sqrt{5}}^{2}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 {(5^{\frac{1}{2}})}^{2}$

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{1}{2} \cdot 2}$

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

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{\frac{2}{2}}$

Divide $1$ by $1$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5^{1}$

Evaluate the exponent.

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$15 - z = 36 - 6 i \sqrt{5} - 6 i \sqrt{5} - 5$

Subtract $5$ from $36$ .

$15 - z = - 6 i \sqrt{5} - 6 i \sqrt{5} + 31$

Subtract $6 i \sqrt{5}$ from $- 6 i \sqrt{5}$ .

$15 - z = - 12 i \sqrt{5} + 31$

Reorder $- 12 i \sqrt{5}$ and $31$ .

$15 - z = 31 - 12 i \sqrt{5}$

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

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Subtract $15$ from both sides of the equation.

$- z = 31 - 12 i \sqrt{5} - 15$

Subtract $15$ from $31$ .

$- z = - 12 i \sqrt{5} + 16$

Reorder $- 12 i \sqrt{5}$ and $16$ .

$- z = 16 - 12 i \sqrt{5}$

Multiply each term in $- z = 16 - 12 i \sqrt{5}$ by $- 1$

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Multiply each term in $- z = 16 - 12 i \sqrt{5}$ by $- 1$ .

$(- z) \cdot - 1 = (16 - 12 i \sqrt{5}) \cdot - 1$

Multiply $(- z) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 z = (16 - 12 i \sqrt{5}) \cdot - 1$

Multiply $z$ by $1$ .

$z = (16 - 12 i \sqrt{5}) \cdot - 1$

Simplify $(16 - 12 i \sqrt{5}) \cdot - 1$ .

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Apply the distributive property.

$z = 16 \cdot - 1 - 12 i \sqrt{5} \cdot - 1$

Multiply.

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Multiply $16$ by $- 1$ .

$z = - 16 - 12 i \sqrt{5} \cdot - 1$

Multiply $- 1$ by $- 12$ .

$z = - 16 + 12 i \sqrt{5}$

Solve for z square root of 15-z+6 = square root of 4-9

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