Solve for a square root of a^2-8a+b^2-6b-25 = square root of a^2+4a+b^2+10b+29

$\sqrt{a^{2} - 8 a + b^{2} - 6 b - 25} = \sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}$

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

${\sqrt{a^{2} - 8 a + b^{2} - 6 b - 25}}^{2} = {\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$

Simplify each side of the equation.

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Multiply the exponents in ${({(a^{2} - 8 a + b^{2} - 6 b - 25)}^{\frac{1}{2}})}^{2}$ .

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

${(a^{2} - 8 a + b^{2} - 6 b - 25)}^{\frac{1}{2} \cdot 2} = {\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

${(a^{2} - 8 a + b^{2} - 6 b - 25)}^{\frac{1}{2} \cdot 2} = {\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$

Rewrite the expression.

${(a^{2} - 8 a + b^{2} - 6 b - 25)}^{1} = {\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$

Simplify.

$a^{2} - 8 a + b^{2} - 6 b - 25 = {\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$

Rewrite ${\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}}^{2}$ as $a^{2} + 4 a + b^{2} + 10 b + 29$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} + 4 a + b^{2} + 10 b + 29}$ as ${(a^{2} + 4 a + b^{2} + 10 b + 29)}^{\frac{1}{2}}$ .

$a^{2} - 8 a + b^{2} - 6 b - 25 = {({(a^{2} + 4 a + b^{2} + 10 b + 29)}^{\frac{1}{2}})}^{2}$

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

$a^{2} - 8 a + b^{2} - 6 b - 25 = {(a^{2} + 4 a + b^{2} + 10 b + 29)}^{\frac{1}{2} \cdot 2}$

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

$a^{2} - 8 a + b^{2} - 6 b - 25 = {(a^{2} + 4 a + b^{2} + 10 b + 29)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$a^{2} - 8 a + b^{2} - 6 b - 25 = {(a^{2} + 4 a + b^{2} + 10 b + 29)}^{\frac{2}{2}}$

Divide $1$ by $1$ .

$a^{2} - 8 a + b^{2} - 6 b - 25 = {(a^{2} + 4 a + b^{2} + 10 b + 29)}^{1}$

Simplify.

$a^{2} - 8 a + b^{2} - 6 b - 25 = a^{2} + 4 a + b^{2} + 10 b + 29$

Solve for $a$ .

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

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Subtract $a^{2}$ from both sides of the equation.

$a^{2} - 8 a + b^{2} - 6 b - 25 - a^{2} = 4 a + b^{2} + 10 b + 29$

Subtract $4 a$ from both sides of the equation.

$a^{2} - 8 a + b^{2} - 6 b - 25 - a^{2} - 4 a = b^{2} + 10 b + 29$

Combine the opposite terms in $a^{2} - 8 a + b^{2} - 6 b - 25 - a^{2} - 4 a$ .

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Subtract $a^{2}$ from $a^{2}$ .

$- 8 a + b^{2} - 6 b - 25 + 0 - 4 a = b^{2} + 10 b + 29$

Add $- 8 a + b^{2} - 6 b - 25$ and $0$ .

$- 8 a + b^{2} - 6 b - 25 - 4 a = b^{2} + 10 b + 29$

Subtract $4 a$ from $- 8 a$ .

$- 12 a + b^{2} - 6 b - 25 = b^{2} + 10 b + 29$

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

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Subtract $b^{2}$ from both sides of the equation.

$- 12 a - 6 b - 25 = b^{2} + 10 b + 29 - b^{2}$

Add $6 b$ to both sides of the equation.

$- 12 a - 25 = b^{2} + 10 b + 29 - b^{2} + 6 b$

Add $25$ to both sides of the equation.

$- 12 a = b^{2} + 10 b + 29 - b^{2} + 6 b + 25$

Combine the opposite terms in $b^{2} + 10 b + 29 - b^{2} + 6 b + 25$ .

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Subtract $b^{2}$ from $b^{2}$ .

$- 12 a = 10 b + 29 + 0 + 6 b + 25$

Add $10 b + 29$ and $0$ .

$- 12 a = 10 b + 29 + 6 b + 25$

Add $10 b$ and $6 b$ .

$- 12 a = 16 b + 29 + 25$

Add $29$ and $25$ .

$- 12 a = 16 b + 54$

Divide each term by $- 12$ and simplify.

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Divide each term in $- 12 a = 16 b + 54$ by $- 12$ .

$\frac{- 12 a}{- 12} = \frac{16 b}{- 12} + \frac{54}{- 12}$

Cancel the common factor of $- 12$ .

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Cancel the common factor.

$\frac{- 12 a}{- 12} = \frac{16 b}{- 12} + \frac{54}{- 12}$

Divide $a$ by $1$ .

$a = \frac{16 b}{- 12} + \frac{54}{- 12}$

Simplify each term.

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Cancel the common factor of $16$ and $- 12$ .

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Factor $4$ out of $16 b$ .

$a = \frac{4 (4 b)}{- 12} + \frac{54}{- 12}$

Cancel the common factors.

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Factor $4$ out of $- 12$ .

$a = \frac{4 (4 b)}{4 (- 3)} + \frac{54}{- 12}$

Cancel the common factor.

$a = \frac{4 (4 b)}{4 \cdot - 3} + \frac{54}{- 12}$

Rewrite the expression.

$a = \frac{4 b}{- 3} + \frac{54}{- 12}$

Move the negative in front of the fraction.

$a = - \frac{4 b}{3} + \frac{54}{- 12}$

Cancel the common factor of $54$ and $- 12$ .

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Factor $6$ out of $54$ .

$a = - \frac{4 b}{3} + \frac{6 (9)}{- 12}$

Cancel the common factors.

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Factor $6$ out of $- 12$ .

$a = - \frac{4 b}{3} + \frac{6 \cdot 9}{6 \cdot - 2}$

Cancel the common factor.

$a = - \frac{4 b}{3} + \frac{6 \cdot 9}{6 \cdot - 2}$

Rewrite the expression.

$a = - \frac{4 b}{3} + \frac{9}{- 2}$

Move the negative in front of the fraction.

$a = - \frac{4 b}{3} - \frac{9}{2}$

Solve for a square root of a^2-8a+b^2-6b-25 = square root of a^2+4a+b^2+10b+29

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