Basic Math Examples

$\frac{5}{1 + R} + \frac{5 + 100}{{(1 + R)}^{2}} = 101.7$

Step 1

Add $5$ and $100$ .

$\frac{5}{1 + R} + \frac{105}{{(1 + R)}^{2}} = 101.7$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$1 + R, {(1 + R)}^{2}, 1$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 2.5

The factor for $1 + R$ is $1 + R$ itself.

$(1 + R) = 1 + R$

$(1 + R)$ occurs $1$ time.

Step 2.6

The factors for $1 + R$ are $(1 + R) \cdot (1 + R)$ , which is $1 + R$ multiplied by itself $2$ times.

$(1 + R) = (1 + R) \cdot (1 + R)$

$(1 + R)$ occurs $2$ times.

Step 2.7

The LCM of $1 + R, {(1 + R)}^{2}$ is the result of multiplying all factors the greatest number of times they occur in either term.

${(1 + R)}^{2}$

Step 3

Multiply each term in $\frac{5}{1 + R} + \frac{105}{{(1 + R)}^{2}} = 101.7$ by ${(1 + R)}^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{5}{1 + R} + \frac{105}{{(1 + R)}^{2}} = 101.7$ by ${(1 + R)}^{2}$ .

$\frac{5}{1 + R} {(1 + R)}^{2} + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $1 + R$ .

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Step 3.2.1.1.1

Factor $1 + R$ out of ${(1 + R)}^{2}$ .

$\frac{5}{1 + R} ((1 + R) (1 + R)) + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.1.2

Cancel the common factor.

$\frac{5}{1 + R} ((1 + R) (1 + R)) + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.1.3

Rewrite the expression.

$5 (1 + R) + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.2

Apply the distributive property.

$5 \cdot 1 + 5 R + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.3

Multiply $5$ by $1$ .

$5 + 5 R + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.4

Cancel the common factor of ${(1 + R)}^{2}$ .

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Step 3.2.1.4.1

Cancel the common factor.

$5 + 5 R + \frac{105}{{(1 + R)}^{2}} {(1 + R)}^{2} = 101.7 {(1 + R)}^{2}$

Step 3.2.1.4.2

Rewrite the expression.

$5 + 5 R + 105 = 101.7 {(1 + R)}^{2}$

Step 3.2.2

Add $5$ and $105$ .

$5 R + 110 = 101.7 {(1 + R)}^{2}$

Step 4

Solve the equation.

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Step 4.1

Simplify $101.7 {(1 + R)}^{2}$ .

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Step 4.1.1

Rewrite.

$5 R + 110 = 0 + 0 + 101.7 {(1 + R)}^{2}$

Step 4.1.2

Rewrite ${(1 + R)}^{2}$ as $(1 + R) (1 + R)$ .

$5 R + 110 = 101.7 ((1 + R) (1 + R))$

Step 4.1.3

Expand $(1 + R) (1 + R)$ using the FOIL Method.

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Step 4.1.3.1

Apply the distributive property.

$5 R + 110 = 101.7 (1 (1 + R) + R (1 + R))$

Step 4.1.3.2

Apply the distributive property.

$5 R + 110 = 101.7 (1 \cdot 1 + 1 R + R (1 + R))$

Step 4.1.3.3

Apply the distributive property.

$5 R + 110 = 101.7 (1 \cdot 1 + 1 R + R \cdot 1 + R \cdot R)$

Step 4.1.4

Simplify and combine like terms.

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Step 4.1.4.1

Simplify each term.

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Step 4.1.4.1.1

Multiply $1$ by $1$ .

$5 R + 110 = 101.7 (1 + 1 R + R \cdot 1 + R \cdot R)$

Step 4.1.4.1.2

Multiply $R$ by $1$ .

$5 R + 110 = 101.7 (1 + R + R \cdot 1 + R \cdot R)$

Step 4.1.4.1.3

Multiply $R$ by $1$ .

$5 R + 110 = 101.7 (1 + R + R + R \cdot R)$

Step 4.1.4.1.4

Multiply $R$ by $R$ .

$5 R + 110 = 101.7 (1 + R + R + R^{2})$

Step 4.1.4.2

Add $R$ and $R$ .

$5 R + 110 = 101.7 (1 + 2 R + R^{2})$

Step 4.1.5

Apply the distributive property.

$5 R + 110 = 101.7 \cdot 1 + 101.7 (2 R) + 101.7 R^{2}$

Step 4.1.6

Simplify.

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Step 4.1.6.1

Multiply $101.7$ by $1$ .

$5 R + 110 = 101.7 + 101.7 (2 R) + 101.7 R^{2}$

Step 4.1.6.2

Multiply $2$ by $101.7$ .

$5 R + 110 = 101.7 + 203.4 R + 101.7 R^{2}$

Step 4.2

Since $R$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$101.7 + 203.4 R + 101.7 R^{2} = 5 R + 110$

Step 4.3

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

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Step 4.3.1

Subtract $5 R$ from both sides of the equation.

$101.7 + 203.4 R + 101.7 R^{2} - 5 R = 110$

Step 4.3.2

Subtract $5 R$ from $203.4 R$ .

$101.7 + 101.7 R^{2} + 198.4 R = 110$

Step 4.4

Subtract $110$ from both sides of the equation.

$101.7 + 101.7 R^{2} + 198.4 R - 110 = 0$

Step 4.5

Subtract $110$ from $101.7$ .

$101.7 R^{2} + 198.4 R - 8.3 = 0$

Step 4.6

Factor $0.1$ out of $101.7 R^{2} + 198.4 R - 8.3$ .

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Step 4.6.1

Factor $0.1$ out of $101.7 R^{2}$ .

$0.1 (1017 R^{2}) + 198.4 R - 8.3 = 0$

Step 4.6.2

Factor $0.1$ out of $198.4 R$ .

$0.1 (1017 R^{2}) + 0.1 (1984 R) - 8.3 = 0$

Step 4.6.3

Factor $0.1$ out of $- 8.3$ .

$0.1 (1017 R^{2}) + 0.1 (1984 R) + 0.1 (- 83) = 0$

Step 4.6.4

Factor $0.1$ out of $0.1 (1017 R^{2}) + 0.1 (1984 R)$ .

$0.1 (1017 R^{2} + 1984 R) + 0.1 (- 83) = 0$

Step 4.6.5

Factor $0.1$ out of $0.1 (1017 R^{2} + 1984 R) + 0.1 (- 83)$ .

$0.1 (1017 R^{2} + 1984 R - 83) = 0$

Step 4.7

Divide each term in $0.1 (1017 R^{2} + 1984 R - 83) = 0$ by $0.1$ and simplify.

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Step 4.7.1

Divide each term in $0.1 (1017 R^{2} + 1984 R - 83) = 0$ by $0.1$ .

$\frac{0.1 (1017 R^{2} + 1984 R - 83)}{0.1} = \frac{0}{0.1}$

Step 4.7.2

Simplify the left side.

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Step 4.7.2.1

Cancel the common factor of $0.1$ .

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Step 4.7.2.1.1

Cancel the common factor.

$\frac{0.1 (1017 R^{2} + 1984 R - 83)}{0.1} = \frac{0}{0.1}$

Step 4.7.2.1.2

Divide $1017 R^{2} + 1984 R - 83$ by $1$ .

$1017 R^{2} + 1984 R - 83 = \frac{0}{0.1}$

Step 4.7.3

Simplify the right side.

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Step 4.7.3.1

Divide $0$ by $0.1$ .

$1017 R^{2} + 1984 R - 83 = 0$

Step 4.8

Use the quadratic formula to find the solutions.

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

Step 4.9

Substitute the values $a = 1017$ , $b = 1984$ , and $c = - 83$ into the quadratic formula and solve for $R$ .

$\frac{- 1984 \pm \sqrt{1984^{2} - 4 \cdot (1017 \cdot - 83)}}{2 \cdot 1017}$

Step 4.10

Simplify.

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Step 4.10.1

Simplify the numerator.

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Step 4.10.1.1

Raise $1984$ to the power of $2$ .

$R = \frac{- 1984 \pm \sqrt{3936256 - 4 \cdot 1017 \cdot - 83}}{2 \cdot 1017}$

Step 4.10.1.2

Multiply $- 4 \cdot 1017 \cdot - 83$ .

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Step 4.10.1.2.1

Multiply $- 4$ by $1017$ .

$R = \frac{- 1984 \pm \sqrt{3936256 - 4068 \cdot - 83}}{2 \cdot 1017}$

Step 4.10.1.2.2

Multiply $- 4068$ by $- 83$ .

$R = \frac{- 1984 \pm \sqrt{3936256 + 337644}}{2 \cdot 1017}$

Step 4.10.1.3

Add $3936256$ and $337644$ .

$R = \frac{- 1984 \pm \sqrt{4273900}}{2 \cdot 1017}$

Step 4.10.1.4

Rewrite $4273900$ as $10^{2} \cdot 42739$ .

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Step 4.10.1.4.1

Factor $100$ out of $4273900$ .

$R = \frac{- 1984 \pm \sqrt{100 (42739)}}{2 \cdot 1017}$

Step 4.10.1.4.2

Rewrite $100$ as $10^{2}$ .

$R = \frac{- 1984 \pm \sqrt{10^{2} \cdot 42739}}{2 \cdot 1017}$

Step 4.10.1.5

Pull terms out from under the radical.

$R = \frac{- 1984 \pm 10 \sqrt{42739}}{2 \cdot 1017}$

Step 4.10.2

Multiply $2$ by $1017$ .

$R = \frac{- 1984 \pm 10 \sqrt{42739}}{2034}$

Step 4.10.3

Simplify $\frac{- 1984 \pm 10 \sqrt{42739}}{2034}$ .

$R = \frac{- 992 \pm 5 \sqrt{42739}}{1017}$

Step 4.11

The final answer is the combination of both solutions.

$R = - \frac{992 - 5 \sqrt{42739}}{1017}, - \frac{992 + 5 \sqrt{42739}}{1017}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$R = - \frac{992 - 5 \sqrt{42739}}{1017}, - \frac{992 + 5 \sqrt{42739}}{1017}$

Decimal Form:

$R = 0.04097408 \dots, - 1.99180987 \dots$