Finite Math Examples

$67416 = 6000 {(1 + r)}^{2}$

Step 1

Subtract $6000 {(1 + r)}^{2}$ from both sides of the equation.

$67416 - 6000 {(1 + r)}^{2} = 0$

Step 2

Simplify $67416 - 6000 {(1 + r)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$67416 - 6000 ((1 + r) (1 + r)) = 0$

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

$67416 - 6000 (1 (1 + r) + r (1 + r)) = 0$

Step 2.1.2.2

Apply the distributive property.

$67416 - 6000 (1 \cdot 1 + 1 r + r (1 + r)) = 0$

Step 2.1.2.3

Apply the distributive property.

$67416 - 6000 (1 \cdot 1 + 1 r + r \cdot 1 + r \cdot r) = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $1$ by $1$ .

$67416 - 6000 (1 + 1 r + r \cdot 1 + r \cdot r) = 0$

Step 2.1.3.1.2

Multiply $r$ by $1$ .

$67416 - 6000 (1 + r + r \cdot 1 + r \cdot r) = 0$

Step 2.1.3.1.3

Multiply $r$ by $1$ .

$67416 - 6000 (1 + r + r + r \cdot r) = 0$

Step 2.1.3.1.4

Multiply $r$ by $r$ .

$67416 - 6000 (1 + r + r + r^{2}) = 0$

Step 2.1.3.2

Add $r$ and $r$ .

$67416 - 6000 (1 + 2 r + r^{2}) = 0$

Step 2.1.4

Apply the distributive property.

$67416 - 6000 \cdot 1 - 6000 (2 r) - 6000 r^{2} = 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $- 6000$ by $1$ .

$67416 - 6000 - 6000 (2 r) - 6000 r^{2} = 0$

Step 2.1.5.2

Multiply $2$ by $- 6000$ .

$67416 - 6000 - 12000 r - 6000 r^{2} = 0$

Step 2.2

Subtract $6000$ from $67416$ .

$61416 - 12000 r - 6000 r^{2} = 0$

Step 3

Factor $- 24$ out of $61416 - 12000 r - 6000 r^{2}$ .

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Step 3.1

Reorder the expression.

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Step 3.1.1

Move $61416$ .

$- 12000 r - 6000 r^{2} + 61416 = 0$

Step 3.1.2

Reorder $- 12000 r$ and $- 6000 r^{2}$ .

$- 6000 r^{2} - 12000 r + 61416 = 0$

Step 3.2

Factor $- 24$ out of $- 6000 r^{2}$ .

$- 24 (250 r^{2}) - 12000 r + 61416 = 0$

Step 3.3

Factor $- 24$ out of $- 12000 r$ .

$- 24 (250 r^{2}) - 24 (500 r) + 61416 = 0$

Step 3.4

Factor $- 24$ out of $61416$ .

$- 24 (250 r^{2}) - 24 (500 r) - 24 \cdot - 2559 = 0$

Step 3.5

Factor $- 24$ out of $- 24 (250 r^{2}) - 24 (500 r)$ .

$- 24 (250 r^{2} + 500 r) - 24 \cdot - 2559 = 0$

Step 3.6

Factor $- 24$ out of $- 24 (250 r^{2} + 500 r) - 24 (- 2559)$ .

$- 24 (250 r^{2} + 500 r - 2559) = 0$

Step 4

Use the quadratic formula to find the roots for $250 r^{2} + 500 r - 2559 = 0$

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Step 4.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} = 0$

Step 4.2

Substitute the values $a = 250$ , $b = 500$ , and $c = - 2559$ into the quadratic formula and solve for $r$ .

$\frac{- 500 \pm \sqrt{500^{2} - 4 \cdot (250 \cdot - 2559)}}{2 \cdot 250} = 0$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the numerator.

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Step 4.3.1.1

Raise $500$ to the power of $2$ .

$r = \frac{- 500 \pm \sqrt{250000 - 4 \cdot 250 \cdot - 2559}}{2 \cdot 250}$

Step 4.3.1.2

Multiply $- 4 \cdot 250 \cdot - 2559$ .

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Step 4.3.1.2.1

Multiply $- 4$ by $250$ .

$r = \frac{- 500 \pm \sqrt{250000 - 1000 \cdot - 2559}}{2 \cdot 250}$

Step 4.3.1.2.2

Multiply $- 1000$ by $- 2559$ .

$r = \frac{- 500 \pm \sqrt{250000 + 2559000}}{2 \cdot 250}$

Step 4.3.1.3

Add $250000$ and $2559000$ .

$r = \frac{- 500 \pm \sqrt{2809000}}{2 \cdot 250}$

Step 4.3.1.4

Rewrite $2809000$ as $530^{2} \cdot 10$ .

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Step 4.3.1.4.1

Factor $280900$ out of $2809000$ .

$r = \frac{- 500 \pm \sqrt{280900 (10)}}{2 \cdot 250}$

Step 4.3.1.4.2

Rewrite $280900$ as $530^{2}$ .

$r = \frac{- 500 \pm \sqrt{530^{2} \cdot 10}}{2 \cdot 250}$

Step 4.3.1.5

Pull terms out from under the radical.

$r = \frac{- 500 \pm 530 \sqrt{10}}{2 \cdot 250}$

Step 4.3.2

Multiply $2$ by $250$ .

$r = \frac{- 500 \pm 530 \sqrt{10}}{500}$

Step 4.3.3

Simplify $\frac{- 500 \pm 530 \sqrt{10}}{500}$ .

$r = \frac{- 50 \pm 53 \sqrt{10}}{50}$

Step 4.4

The final answer is the combination of both solutions.

$r = - \frac{50 - 53 \sqrt{10}}{50}, - \frac{50 + 53 \sqrt{10}}{50}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{50 - 53 \sqrt{10}}{50}, - \frac{50 + 53 \sqrt{10}}{50}$

Decimal Form:

$r = 2.35201431 \dots, - 4.35201431 \dots$