Algebra Examples

$\sqrt{29 + 4 w} = 23 - c w$

Step 1

Rewrite the equation as $23 - c w = \sqrt{29 + 4 w}$ .

$23 - c w = \sqrt{29 + 4 w}$

Step 2

Subtract $23$ from both sides of the equation.

$- c w = \sqrt{29 + 4 w} - 23$

Step 3

Divide each term in $- c w = \sqrt{29 + 4 w} - 23$ by $- w$ and simplify.

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Step 3.1

Divide each term in $- c w = \sqrt{29 + 4 w} - 23$ by $- w$ .

$\frac{- c w}{- w} = \frac{\sqrt{29 + 4 w}}{- w} + \frac{- 23}{- w}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{c w}{w} = \frac{\sqrt{29 + 4 w}}{- w} + \frac{- 23}{- w}$

Step 3.2.2

Cancel the common factor of $w$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{c w}{w} = \frac{\sqrt{29 + 4 w}}{- w} + \frac{- 23}{- w}$

Step 3.2.2.2

Divide $c$ by $1$ .

$c = \frac{\sqrt{29 + 4 w}}{- w} + \frac{- 23}{- w}$

Step 3.3

Simplify the right side.

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Step 3.3.1

To write $\frac{\sqrt{29 + 4 w}}{- w}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$c = \frac{\sqrt{29 + 4 w}}{- w} \cdot \frac{- 1}{- 1} + \frac{- 23}{- w}$

Step 3.3.2

To write $\frac{- 23}{- w}$ as a fraction with a common denominator, multiply by $\frac{- 1}{- 1}$ .

$c = \frac{\sqrt{29 + 4 w}}{- w} \cdot \frac{- 1}{- 1} + \frac{- 23}{- w} \cdot \frac{- 1}{- 1}$

Step 3.3.3

Write each expression with a common denominator of $w$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.3.1

Multiply $\frac{\sqrt{29 + 4 w}}{- w}$ by $\frac{- 1}{- 1}$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{- w \cdot - 1} + \frac{- 23}{- w} \cdot \frac{- 1}{- 1}$

Step 3.3.3.2

Multiply $- 1$ by $- 1$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{1 w} + \frac{- 23}{- w} \cdot \frac{- 1}{- 1}$

Step 3.3.3.3

Multiply $w$ by $1$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{w} + \frac{- 23}{- w} \cdot \frac{- 1}{- 1}$

Step 3.3.3.4

Multiply $\frac{- 23}{- w}$ by $\frac{- 1}{- 1}$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{w} + \frac{- 23 \cdot - 1}{- w \cdot - 1}$

Step 3.3.3.5

Multiply $- 1$ by $- 1$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{w} + \frac{- 23 \cdot - 1}{1 w}$

Step 3.3.3.6

Multiply $w$ by $1$ .

$c = \frac{\sqrt{29 + 4 w} \cdot - 1}{w} + \frac{- 23 \cdot - 1}{w}$

Step 3.3.4

Combine the numerators over the common denominator.

$c = \frac{\sqrt{29 + 4 w} \cdot - 1 - 23 \cdot - 1}{w}$

Step 3.3.5

Simplify each term.

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Step 3.3.5.1

Move $- 1$ to the left of $\sqrt{29 + 4 w}$ .

$c = \frac{- 1 \cdot \sqrt{29 + 4 w} - 23 \cdot - 1}{w}$

Step 3.3.5.2

Rewrite $- 1 \sqrt{29 + 4 w}$ as $- \sqrt{29 + 4 w}$ .

$c = \frac{- \sqrt{29 + 4 w} - 23 \cdot - 1}{w}$

Step 3.3.5.3

Multiply $- 23$ by $- 1$ .

$c = \frac{- \sqrt{29 + 4 w} + 23}{w}$

Step 3.3.6

Simplify with factoring out.

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Step 3.3.6.1

Factor $- 1$ out of $- \sqrt{29 + 4 w}$ .

$c = \frac{- (\sqrt{29 + 4 w}) + 23}{w}$

Step 3.3.6.2

Rewrite $23$ as $- 1 (- 23)$ .

$c = \frac{- (\sqrt{29 + 4 w}) - 1 (- 23)}{w}$

Step 3.3.6.3

Factor $- 1$ out of $- (\sqrt{29 + 4 w}) - 1 (- 23)$ .

$c = \frac{- (\sqrt{29 + 4 w} - 23)}{w}$

Step 3.3.6.4

Simplify the expression.

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Step 3.3.6.4.1

Rewrite $- (\sqrt{29 + 4 w} - 23)$ as $- 1 (\sqrt{29 + 4 w} - 23)$ .

$c = \frac{- 1 (\sqrt{29 + 4 w} - 23)}{w}$

Step 3.3.6.4.2

Move the negative in front of the fraction.

$c = - \frac{\sqrt{29 + 4 w} - 23}{w}$