Algebra Examples

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} = 15$

Step 1

Multiply both sides by $R_{1} + 2 \sqrt{R_{1}}$ .

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} (R_{1} + 2 \sqrt{R_{1}}) = 15 (R_{1} + 2 \sqrt{R_{1}})$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Cancel the common factor of $R_{1} + 2 \sqrt{R_{1}}$ .

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Step 2.1.1.1

Cancel the common factor.

$\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} (R_{1} + 2 \sqrt{R_{1}}) = 15 (R_{1} + 2 \sqrt{R_{1}})$

Step 2.1.1.2

Rewrite the expression.

$2 R_{1} \sqrt{R_{1}} = 15 (R_{1} + 2 \sqrt{R_{1}})$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $15 (R_{1} + 2 \sqrt{R_{1}})$ .

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Step 2.2.1.1

Apply the distributive property.

$2 R_{1} \sqrt{R_{1}} = 15 R_{1} + 15 (2 \sqrt{R_{1}})$

Step 2.2.1.2

Multiply $2$ by $15$ .

$2 R_{1} \sqrt{R_{1}} = 15 R_{1} + 30 \sqrt{R_{1}}$

Step 3

Solve for $R_{1}$ .

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Step 3.1

Solve for $\sqrt{R_{1}}$ .

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Step 3.1.1

Subtract $30 \sqrt{R_{1}}$ from both sides of the equation.

$2 R_{1} \sqrt{R_{1}} - 30 \sqrt{R_{1}} = 15 R_{1}$

Step 3.1.2

Factor $2 \sqrt{R_{1}}$ out of $2 R_{1} \sqrt{R_{1}} - 30 \sqrt{R_{1}}$ .

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Step 3.1.2.1

Factor $2 \sqrt{R_{1}}$ out of $2 R_{1} \sqrt{R_{1}}$ .

$2 \sqrt{R_{1}} R_{1} - 30 \sqrt{R_{1}} = 15 R_{1}$

Step 3.1.2.2

Factor $2 \sqrt{R_{1}}$ out of $- 30 \sqrt{R_{1}}$ .

$2 \sqrt{R_{1}} R_{1} + 2 \sqrt{R_{1}} \cdot - 15 = 15 R_{1}$

Step 3.1.2.3

Factor $2 \sqrt{R_{1}}$ out of $2 \sqrt{R_{1}} R_{1} + 2 \sqrt{R_{1}} \cdot - 15$ .

$2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$

Step 3.1.3

Divide each term in $2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$ by $2 (R_{1} - 15)$ and simplify.

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Step 3.1.3.1

Divide each term in $2 \sqrt{R_{1}} (R_{1} - 15) = 15 R_{1}$ by $2 (R_{1} - 15)$ .

$\frac{2 \sqrt{R_{1}} (R_{1} - 15)}{2 (R_{1} - 15)} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Step 3.1.3.2

Simplify the left side.

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Step 3.1.3.2.1

Cancel the common factor of $2$ .

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Step 3.1.3.2.1.1

Cancel the common factor.

$\frac{2 \sqrt{R_{1}} (R_{1} - 15)}{2 (R_{1} - 15)} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Step 3.1.3.2.1.2

Rewrite the expression.

$\frac{\sqrt{R_{1}} (R_{1} - 15)}{R_{1} - 15} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Step 3.1.3.2.2

Cancel the common factor of $R_{1} - 15$ .

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Step 3.1.3.2.2.1

Cancel the common factor.

$\frac{\sqrt{R_{1}} (R_{1} - 15)}{R_{1} - 15} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Step 3.1.3.2.2.2

Divide $\sqrt{R_{1}}$ by $1$ .

$\sqrt{R_{1}} = \frac{15 R_{1}}{2 (R_{1} - 15)}$

Step 3.2

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

${\sqrt{R_{1}}}^{2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

${({R_{1}}^{\frac{1}{2}})}^{2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${({R_{1}}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Multiply the exponents in ${({R_{1}}^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1.1

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

${R_{1}}^{\frac{1}{2} \cdot 2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.1.2.1

Cancel the common factor.

${R_{1}}^{\frac{1}{2} \cdot 2} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3.2.1.1.2.2

Rewrite the expression.

${R_{1}}^{1} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3.2.1.2

Simplify.

$R_{1} = {(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify ${(\frac{15 R_{1}}{2 (R_{1} - 15)})}^{2}$ .

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Step 3.3.3.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.3.1.1.1

Apply the product rule to $\frac{15 R_{1}}{2 (R_{1} - 15)}$ .

$R_{1} = \frac{{(15 R_{1})}^{2}}{{(2 (R_{1} - 15))}^{2}}$

Step 3.3.3.1.1.2

Apply the product rule to $15 R_{1}$ .

$R_{1} = \frac{15^{2} {R_{1}}^{2}}{{(2 (R_{1} - 15))}^{2}}$

Step 3.3.3.1.1.3

Apply the product rule to $2 (R_{1} - 15)$ .

$R_{1} = \frac{15^{2} {R_{1}}^{2}}{2^{2} {(R_{1} - 15)}^{2}}$

Step 3.3.3.1.2

Raise $15$ to the power of $2$ .

$R_{1} = \frac{225 {R_{1}}^{2}}{2^{2} {(R_{1} - 15)}^{2}}$

Step 3.3.3.1.3

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

$R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$

Step 3.4

Solve for $R_{1}$ .

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Step 3.4.1

Find the LCD of the terms in the equation.

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Step 3.4.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 4 {(R_{1} - 15)}^{2}$

Step 3.4.1.2

The LCM of one and any expression is the expression.

$4 {(R_{1} - 15)}^{2}$

Step 3.4.2

Multiply each term in $R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$ by $4 {(R_{1} - 15)}^{2}$ to eliminate the fractions.

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Step 3.4.2.1

Multiply each term in $R_{1} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}}$ by $4 {(R_{1} - 15)}^{2}$ .

$R_{1} (4 {(R_{1} - 15)}^{2}) = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} (4 {(R_{1} - 15)}^{2})$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Rewrite using the commutative property of multiplication.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} (4 {(R_{1} - 15)}^{2})$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Rewrite using the commutative property of multiplication.

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Step 3.4.2.3.2

Cancel the common factor of $4$ .

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Step 3.4.2.3.2.1

Factor $4$ out of $4 {(R_{1} - 15)}^{2}$ .

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Step 3.4.2.3.2.2

Cancel the common factor.

$4 R_{1} {(R_{1} - 15)}^{2} = 4 \frac{225 {R_{1}}^{2}}{4 {(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Step 3.4.2.3.2.3

Rewrite the expression.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{{(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Step 3.4.2.3.3

Cancel the common factor of ${(R_{1} - 15)}^{2}$ .

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Step 3.4.2.3.3.1

Cancel the common factor.

$4 R_{1} {(R_{1} - 15)}^{2} = \frac{225 {R_{1}}^{2}}{{(R_{1} - 15)}^{2}} {(R_{1} - 15)}^{2}$

Step 3.4.2.3.3.2

Rewrite the expression.

$4 R_{1} {(R_{1} - 15)}^{2} = 225 {R_{1}}^{2}$

Step 3.4.3

Solve the equation.

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Step 3.4.3.1

Move all terms containing $R_{1}$ to the left side of the equation.

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Step 3.4.3.1.1

Subtract $225 {R_{1}}^{2}$ from both sides of the equation.

$4 R_{1} {(R_{1} - 15)}^{2} - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2

Simplify each term.

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Step 3.4.3.1.2.1

Rewrite ${(R_{1} - 15)}^{2}$ as $(R_{1} - 15) (R_{1} - 15)$ .

$4 R_{1} ((R_{1} - 15) (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.2

Expand $(R_{1} - 15) (R_{1} - 15)$ using the FOIL Method.

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Step 3.4.3.1.2.2.1

Apply the distributive property.

$4 R_{1} (R_{1} (R_{1} - 15) - 15 (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.2.2

Apply the distributive property.

$4 R_{1} (R_{1} R_{1} + R_{1} \cdot - 15 - 15 (R_{1} - 15)) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.2.3

Apply the distributive property.

$4 R_{1} (R_{1} R_{1} + R_{1} \cdot - 15 - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.3

Simplify and combine like terms.

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Step 3.4.3.1.2.3.1

Simplify each term.

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Step 3.4.3.1.2.3.1.1

Multiply $R_{1}$ by $R_{1}$ .

$4 R_{1} ({R_{1}}^{2} + R_{1} \cdot - 15 - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.3.1.2

Move $- 15$ to the left of $R_{1}$ .

$4 R_{1} ({R_{1}}^{2} - 15 \cdot R_{1} - 15 R_{1} - 15 \cdot - 15) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.3.1.3

Multiply $- 15$ by $- 15$ .

$4 R_{1} ({R_{1}}^{2} - 15 R_{1} - 15 R_{1} + 225) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.3.2

Subtract $15 R_{1}$ from $- 15 R_{1}$ .

$4 R_{1} ({R_{1}}^{2} - 30 R_{1} + 225) - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.4

Apply the distributive property.

$4 R_{1} {R_{1}}^{2} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5

Simplify.

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Step 3.4.3.1.2.5.1

Multiply $R_{1}$ by ${R_{1}}^{2}$ by adding the exponents.

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Step 3.4.3.1.2.5.1.1

Move ${R_{1}}^{2}$ .

$4 ({R_{1}}^{2} R_{1}) + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5.1.2

Multiply ${R_{1}}^{2}$ by $R_{1}$ .

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Step 3.4.3.1.2.5.1.2.1

Raise $R_{1}$ to the power of $1$ .

$4 ({R_{1}}^{2} {R_{1}}^{1}) + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5.1.2.2

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

$4 {R_{1}}^{2 + 1} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5.1.3

Add $2$ and $1$ .

$4 {R_{1}}^{3} + 4 R_{1} (- 30 R_{1}) + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5.2

Rewrite using the commutative property of multiplication.

$4 {R_{1}}^{3} + 4 \cdot - 30 R_{1} R_{1} + 4 R_{1} \cdot 225 - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.5.3

Multiply $225$ by $4$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 R_{1} R_{1} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.6

Simplify each term.

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Step 3.4.3.1.2.6.1

Multiply $R_{1}$ by $R_{1}$ by adding the exponents.

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Step 3.4.3.1.2.6.1.1

Move $R_{1}$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 (R_{1} R_{1}) + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.6.1.2

Multiply $R_{1}$ by $R_{1}$ .

$4 {R_{1}}^{3} + 4 \cdot - 30 {R_{1}}^{2} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.2.6.2

Multiply $4$ by $- 30$ .

$4 {R_{1}}^{3} - 120 {R_{1}}^{2} + 900 R_{1} - 225 {R_{1}}^{2} = 0$

Step 3.4.3.1.3

Subtract $225 {R_{1}}^{2}$ from $- 120 {R_{1}}^{2}$ .

$4 {R_{1}}^{3} - 345 {R_{1}}^{2} + 900 R_{1} = 0$

Step 3.4.3.2

Factor $R_{1}$ out of $4 {R_{1}}^{3} - 345 {R_{1}}^{2} + 900 R_{1}$ .

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Step 3.4.3.2.1

Factor $R_{1}$ out of $4 {R_{1}}^{3}$ .

$R_{1} (4 {R_{1}}^{2}) - 345 {R_{1}}^{2} + 900 R_{1} = 0$

Step 3.4.3.2.2

Factor $R_{1}$ out of $- 345 {R_{1}}^{2}$ .

$R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1}) + 900 R_{1} = 0$

Step 3.4.3.2.3

Factor $R_{1}$ out of $900 R_{1}$ .

$R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1}) + R_{1} \cdot 900 = 0$

Step 3.4.3.2.4

Factor $R_{1}$ out of $R_{1} (4 {R_{1}}^{2}) + R_{1} (- 345 R_{1})$ .

$R_{1} (4 {R_{1}}^{2} - 345 R_{1}) + R_{1} \cdot 900 = 0$

Step 3.4.3.2.5

Factor $R_{1}$ out of $R_{1} (4 {R_{1}}^{2} - 345 R_{1}) + R_{1} \cdot 900$ .

$R_{1} (4 {R_{1}}^{2} - 345 R_{1} + 900) = 0$

Step 3.4.3.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$R_{1} = 0$

$4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$

Step 3.4.3.4

Set $R_{1}$ equal to $0$ .

$R_{1} = 0$

Step 3.4.3.5

Set $4 {R_{1}}^{2} - 345 R_{1} + 900$ equal to $0$ and solve for $R_{1}$ .

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Step 3.4.3.5.1

Set $4 {R_{1}}^{2} - 345 R_{1} + 900$ equal to $0$ .

$4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$

Step 3.4.3.5.2

Solve $4 {R_{1}}^{2} - 345 R_{1} + 900 = 0$ for $R_{1}$ .

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Step 3.4.3.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.4.3.5.2.2

Substitute the values $a = 4$ , $b = - 345$ , and $c = 900$ into the quadratic formula and solve for $R_{1}$ .

$\frac{345 \pm \sqrt{{(- 345)}^{2} - 4 \cdot (4 \cdot 900)}}{2 \cdot 4}$

Step 3.4.3.5.2.3

Simplify.

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Step 3.4.3.5.2.3.1

Simplify the numerator.

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Step 3.4.3.5.2.3.1.1

Raise $- 345$ to the power of $2$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 4 \cdot 4 \cdot 900}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.2

Multiply $- 4 \cdot 4 \cdot 900$ .

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Step 3.4.3.5.2.3.1.2.1

Multiply $- 4$ by $4$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 16 \cdot 900}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.2.2

Multiply $- 16$ by $900$ .

$R_{1} = \frac{345 \pm \sqrt{119025 - 14400}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.3

Subtract $14400$ from $119025$ .

$R_{1} = \frac{345 \pm \sqrt{104625}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.4

Rewrite $104625$ as $15^{2} \cdot 465$ .

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Step 3.4.3.5.2.3.1.4.1

Factor $225$ out of $104625$ .

$R_{1} = \frac{345 \pm \sqrt{225 (465)}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.4.2

Rewrite $225$ as $15^{2}$ .

$R_{1} = \frac{345 \pm \sqrt{15^{2} \cdot 465}}{2 \cdot 4}$

Step 3.4.3.5.2.3.1.5

Pull terms out from under the radical.

$R_{1} = \frac{345 \pm 15 \sqrt{465}}{2 \cdot 4}$

Step 3.4.3.5.2.3.2

Multiply $2$ by $4$ .

$R_{1} = \frac{345 \pm 15 \sqrt{465}}{8}$

Step 3.4.3.5.2.4

The final answer is the combination of both solutions.

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}, \frac{345 - 15 \sqrt{465}}{8}$

Step 3.4.3.6

The final solution is all the values that make $R_{1} (4 {R_{1}}^{2} - 345 R_{1} + 900) = 0$ true.

$R_{1} = 0, \frac{345 + 15 \sqrt{465}}{8}, \frac{345 - 15 \sqrt{465}}{8}$

Step 4

Exclude the solutions that do not make $\frac{2 R_{1} \sqrt{R_{1}}}{R_{1} + 2 \sqrt{R_{1}}} = 15$ true.

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$R_{1} = \frac{345 + 15 \sqrt{465}}{8}$

Decimal Form:

$R_{1} = 83.55723497 \dots$