Finite Math Examples

$5565.13 = 5000 {(1 + r)}^{2}$

Step 1

Subtract $5000 {(1 + r)}^{2}$ from both sides of the equation.

$5565.13 - 5000 {(1 + r)}^{2} = 0$

Step 2

Simplify $5565.13 - 5000 {(1 + r)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$5565.13 - 5000 ((1 + r) (1 + r)) = 0$

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.

$5565.13 - 5000 (1 (1 + r) + r (1 + r)) = 0$

Step 2.1.2.2

Apply the distributive property.

$5565.13 - 5000 (1 \cdot 1 + 1 r + r (1 + r)) = 0$

Step 2.1.2.3

Apply the distributive property.

$5565.13 - 5000 (1 \cdot 1 + 1 r + r \cdot 1 + r \cdot r) = 0$

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 $1$ .

$5565.13 - 5000 (1 + 1 r + r \cdot 1 + r \cdot r) = 0$

Step 2.1.3.1.2

Multiply $r$ by $1$ .

$5565.13 - 5000 (1 + r + r \cdot 1 + r \cdot r) = 0$

Step 2.1.3.1.3

Multiply $r$ by $1$ .

$5565.13 - 5000 (1 + r + r + r \cdot r) = 0$

Step 2.1.3.1.4

Multiply $r$ by $r$ .

$5565.13 - 5000 (1 + r + r + r^{2}) = 0$

Step 2.1.3.2

Add $r$ and $r$ .

$5565.13 - 5000 (1 + 2 r + r^{2}) = 0$

Step 2.1.4

Apply the distributive property.

$5565.13 - 5000 \cdot 1 - 5000 (2 r) - 5000 r^{2} = 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $- 5000$ by $1$ .

$5565.13 - 5000 - 5000 (2 r) - 5000 r^{2} = 0$

Step 2.1.5.2

Multiply $2$ by $- 5000$ .

$5565.13 - 5000 - 10000 r - 5000 r^{2} = 0$

Step 2.2

Subtract $5000$ from $5565.13$ .

$565.13 - 10000 r - 5000 r^{2} = 0$

Step 3

Factor $0.01$ out of $565.13 - 10000 r - 5000 r^{2}$ .

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Step 3.1

Factor $0.01$ out of $565.13$ .

$0.01 (56513) - 10000 r - 5000 r^{2} = 0$

Step 3.2

Factor $0.01$ out of $- 10000 r$ .

$0.01 (56513) + 0.01 (- 1000000 r) - 5000 r^{2} = 0$

Step 3.3

Factor $0.01$ out of $- 5000 r^{2}$ .

$0.01 (56513) + 0.01 (- 1000000 r) + 0.01 (- 500000 r^{2}) = 0$

Step 3.4

Factor $0.01$ out of $0.01 (56513) + 0.01 (- 1000000 r)$ .

$0.01 (56513 - 1000000 r) + 0.01 (- 500000 r^{2}) = 0$

Step 3.5

Factor $0.01$ out of $0.01 (56513 - 1000000 r) + 0.01 (- 500000 r^{2})$ .

$0.01 (56513 - 1000000 r - 500000 r^{2}) = 0$

Step 4

Divide each term in $0.01 (56513 - 1000000 r - 500000 r^{2}) = 0$ by $0.01$ and simplify.

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Step 4.1

Divide each term in $0.01 (56513 - 1000000 r - 500000 r^{2}) = 0$ by $0.01$ .

$\frac{0.01 (56513 - 1000000 r - 500000 r^{2})}{0.01} = \frac{0}{0.01}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.01$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.01 (56513 - 1000000 r - 500000 r^{2})}{0.01} = \frac{0}{0.01}$

Step 4.2.1.2

Divide $56513 - 1000000 r - 500000 r^{2}$ by $1$ .

$56513 - 1000000 r - 500000 r^{2} = \frac{0}{0.01}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.01$ .

$56513 - 1000000 r - 500000 r^{2} = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 500000$ , $b = - 1000000$ , and $c = 56513$ into the quadratic formula and solve for $r$ .

$\frac{1000000 \pm \sqrt{{(- 1000000)}^{2} - 4 \cdot (- 500000 \cdot 56513)}}{2 \cdot - 500000}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 1000000$ to the power of $2$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 - 4 \cdot - 500000 \cdot 56513}}{2 \cdot - 500000}$

Step 7.1.2

Multiply $- 4 \cdot - 500000 \cdot 56513$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 500000$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 2000000 \cdot 56513}}{2 \cdot - 500000}$

Step 7.1.2.2

Multiply $2000000$ by $56513$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 113026000000}}{2 \cdot - 500000}$

Step 7.1.3

Add $1000000000000$ and $113026000000$ .

$r = \frac{1000000 \pm \sqrt{1113026000000}}{2 \cdot - 500000}$

Step 7.1.4

Rewrite $1113026000000$ as $1000^{2} \cdot 1113026$ .

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Step 7.1.4.1

Factor $1000000$ out of $1113026000000$ .

$r = \frac{1000000 \pm \sqrt{1000000 (1113026)}}{2 \cdot - 500000}$

Step 7.1.4.2

Rewrite $1000000$ as $1000^{2}$ .

$r = \frac{1000000 \pm \sqrt{1000^{2} \cdot 1113026}}{2 \cdot - 500000}$

Step 7.1.5

Pull terms out from under the radical.

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{2 \cdot - 500000}$

Step 7.2

Multiply $2$ by $- 500000$ .

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$

Step 7.3

Simplify $\frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$ .

$r = \frac{1000 \pm \sqrt{1113026}}{- 1000}$

Step 7.4

Move the negative in front of the fraction.

$r = - \frac{1000 \pm \sqrt{1113026}}{1000}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 1000000$ to the power of $2$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 - 4 \cdot - 500000 \cdot 56513}}{2 \cdot - 500000}$

Step 8.1.2

Multiply $- 4 \cdot - 500000 \cdot 56513$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 500000$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 2000000 \cdot 56513}}{2 \cdot - 500000}$

Step 8.1.2.2

Multiply $2000000$ by $56513$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 113026000000}}{2 \cdot - 500000}$

Step 8.1.3

Add $1000000000000$ and $113026000000$ .

$r = \frac{1000000 \pm \sqrt{1113026000000}}{2 \cdot - 500000}$

Step 8.1.4

Rewrite $1113026000000$ as $1000^{2} \cdot 1113026$ .

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Step 8.1.4.1

Factor $1000000$ out of $1113026000000$ .

$r = \frac{1000000 \pm \sqrt{1000000 (1113026)}}{2 \cdot - 500000}$

Step 8.1.4.2

Rewrite $1000000$ as $1000^{2}$ .

$r = \frac{1000000 \pm \sqrt{1000^{2} \cdot 1113026}}{2 \cdot - 500000}$

Step 8.1.5

Pull terms out from under the radical.

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{2 \cdot - 500000}$

Step 8.2

Multiply $2$ by $- 500000$ .

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$

Step 8.3

Simplify $\frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$ .

$r = \frac{1000 \pm \sqrt{1113026}}{- 1000}$

Step 8.4

Move the negative in front of the fraction.

$r = - \frac{1000 \pm \sqrt{1113026}}{1000}$

Step 8.5

Change the $\pm$ to $+$ .

$r = - \frac{1000 + \sqrt{1113026}}{1000}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 1000000$ to the power of $2$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 - 4 \cdot - 500000 \cdot 56513}}{2 \cdot - 500000}$

Step 9.1.2

Multiply $- 4 \cdot - 500000 \cdot 56513$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 500000$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 2000000 \cdot 56513}}{2 \cdot - 500000}$

Step 9.1.2.2

Multiply $2000000$ by $56513$ .

$r = \frac{1000000 \pm \sqrt{1000000000000 + 113026000000}}{2 \cdot - 500000}$

Step 9.1.3

Add $1000000000000$ and $113026000000$ .

$r = \frac{1000000 \pm \sqrt{1113026000000}}{2 \cdot - 500000}$

Step 9.1.4

Rewrite $1113026000000$ as $1000^{2} \cdot 1113026$ .

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Step 9.1.4.1

Factor $1000000$ out of $1113026000000$ .

$r = \frac{1000000 \pm \sqrt{1000000 (1113026)}}{2 \cdot - 500000}$

Step 9.1.4.2

Rewrite $1000000$ as $1000^{2}$ .

$r = \frac{1000000 \pm \sqrt{1000^{2} \cdot 1113026}}{2 \cdot - 500000}$

Step 9.1.5

Pull terms out from under the radical.

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{2 \cdot - 500000}$

Step 9.2

Multiply $2$ by $- 500000$ .

$r = \frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$

Step 9.3

Simplify $\frac{1000000 \pm 1000 \sqrt{1113026}}{- 1000000}$ .

$r = \frac{1000 \pm \sqrt{1113026}}{- 1000}$

Step 9.4

Move the negative in front of the fraction.

$r = - \frac{1000 \pm \sqrt{1113026}}{1000}$

Step 9.5

Change the $\pm$ to $-$ .

$r = - \frac{1000 - \sqrt{1113026}}{1000}$

Step 10

The final answer is the combination of both solutions.

$r = - \frac{1000 + \sqrt{1113026}}{1000}, - \frac{1000 - \sqrt{1113026}}{1000}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{1000 + \sqrt{1113026}}{1000}, - \frac{1000 - \sqrt{1113026}}{1000}$

Decimal Form:

$r = - 2.05500047 \dots, 0.05500047 \dots$