Basic Math Examples

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

Step 1

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

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

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{2} - 8 a + b^{2} - 6 b - 25}$ as ${(a^{2} - 8 a + b^{2} - 6 b - 25)}^{\frac{1}{2}}$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(a^{2} - 8 a + b^{2} - 6 b - 25)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

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

Step 2.2.1.1.2.2

Rewrite the expression.

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

Step 2.2.1.2

Simplify.

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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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Step 2.3.1.1

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

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Cancel the common factor.

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

Step 2.3.1.4.2

Rewrite the expression.

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

Step 2.3.1.5

Simplify.

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

Step 3

Solve for $a$ .

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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Step 3.1.3.1

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

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

Step 3.1.3.2

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

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

Step 3.1.4

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Add $25$ to both sides of the equation.

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

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

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

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

Step 3.2.5

Add $10 b$ and $6 b$ .

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

Step 3.2.6

Add $29$ and $25$ .

$- 12 a = 16 b + 54$

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 12$ .

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Step 3.3.2.1.1

Cancel the common factor.

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

Step 3.3.2.1.2

Divide $a$ by $1$ .

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

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

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

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Step 3.3.3.1.1.1

Factor $4$ out of $16 b$ .

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

Step 3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.1.1.2.1

Factor $4$ out of $- 12$ .

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

Step 3.3.3.1.1.2.2

Cancel the common factor.

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

Step 3.3.3.1.1.2.3

Rewrite the expression.

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

Step 3.3.3.1.2

Move the negative in front of the fraction.

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

Step 3.3.3.1.3

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

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Step 3.3.3.1.3.1

Factor $6$ out of $54$ .

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

Step 3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.1.3.2.1

Factor $6$ out of $- 12$ .

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

Step 3.3.3.1.3.2.2

Cancel the common factor.

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

Step 3.3.3.1.3.2.3

Rewrite the expression.

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

Step 3.3.3.1.4

Move the negative in front of the fraction.

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