Finite Math Examples

$16000 {(1 + r)}^{2} = 25000$

Step 1

Subtract $25000$ from both sides of the equation.

$16000 {(1 + r)}^{2} - 25000 = 0$

Step 2

Simplify $16000 {(1 + r)}^{2} - 25000$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$16000 ((1 + r) (1 + r)) - 25000 = 0$

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.

$16000 (1 (1 + r) + r (1 + r)) - 25000 = 0$

Step 2.1.2.2

Apply the distributive property.

$16000 (1 \cdot 1 + 1 r + r (1 + r)) - 25000 = 0$

Step 2.1.2.3

Apply the distributive property.

$16000 (1 \cdot 1 + 1 r + r \cdot 1 + r \cdot r) - 25000 = 0$

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 $1$ .

$16000 (1 + 1 r + r \cdot 1 + r \cdot r) - 25000 = 0$

Step 2.1.3.1.2

Multiply $r$ by $1$ .

$16000 (1 + r + r \cdot 1 + r \cdot r) - 25000 = 0$

Step 2.1.3.1.3

Multiply $r$ by $1$ .

$16000 (1 + r + r + r \cdot r) - 25000 = 0$

Step 2.1.3.1.4

Multiply $r$ by $r$ .

$16000 (1 + r + r + r^{2}) - 25000 = 0$

Step 2.1.3.2

Add $r$ and $r$ .

$16000 (1 + 2 r + r^{2}) - 25000 = 0$

Step 2.1.4

Apply the distributive property.

$16000 \cdot 1 + 16000 (2 r) + 16000 r^{2} - 25000 = 0$

Step 2.1.5

Simplify.

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Step 2.1.5.1

Multiply $16000$ by $1$ .

$16000 + 16000 (2 r) + 16000 r^{2} - 25000 = 0$

Step 2.1.5.2

Multiply $2$ by $16000$ .

$16000 + 32000 r + 16000 r^{2} - 25000 = 0$

Step 2.2

Subtract $25000$ from $16000$ .

$32000 r + 16000 r^{2} - 9000 = 0$

Step 3

Factor $1000$ out of $32000 r + 16000 r^{2} - 9000$ .

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Step 3.1

Factor $1000$ out of $32000 r$ .

$1000 (32 r) + 16000 r^{2} - 9000 = 0$

Step 3.2

Factor $1000$ out of $16000 r^{2}$ .

$1000 (32 r) + 1000 (16 r^{2}) - 9000 = 0$

Step 3.3

Factor $1000$ out of $- 9000$ .

$1000 (32 r) + 1000 (16 r^{2}) + 1000 (- 9) = 0$

Step 3.4

Factor $1000$ out of $1000 (32 r) + 1000 (16 r^{2})$ .

$1000 (32 r + 16 r^{2}) + 1000 (- 9) = 0$

Step 3.5

Factor $1000$ out of $1000 (32 r + 16 r^{2}) + 1000 (- 9)$ .

$1000 (32 r + 16 r^{2} - 9) = 0$

Step 4

Let $u = r$ . Substitute $u$ for all occurrences of $r$ .

$1000 (32 u + 16 u^{2} - 9) = 0$

Step 5

Factor by grouping.

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Step 5.1

Reorder terms.

$1000 (16 u^{2} + 32 u - 9) = 0$

Step 5.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 16 \cdot - 9 = - 144$ and whose sum is $b = 32$ .

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Step 5.2.1

Factor $32$ out of $32 u$ .

$1000 (16 u^{2} + 32 (u) - 9) = 0$

Step 5.2.2

Rewrite $32$ as $- 4$ plus $36$

$1000 (16 u^{2} + (- 4 + 36) u - 9) = 0$

Step 5.2.3

Apply the distributive property.

$1000 (16 u^{2} - 4 u + 36 u - 9) = 0$

Step 5.3

Factor out the greatest common factor from each group.

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Step 5.3.1

Group the first two terms and the last two terms.

$1000 ((16 u^{2} - 4 u) + 36 u - 9) = 0$

Step 5.3.2

Factor out the greatest common factor (GCF) from each group.

$1000 (4 u (4 u - 1) + 9 (4 u - 1)) = 0$

Step 5.4

Factor the polynomial by factoring out the greatest common factor, $4 u - 1$ .

$1000 ((4 u - 1) (4 u + 9)) = 0$

Step 6

Factor.

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Step 6.1

Replace all occurrences of $u$ with $r$ .

$1000 ((4 r - 1) (4 r + 9)) = 0$

Step 6.2

Remove unnecessary parentheses.

$1000 (4 r - 1) (4 r + 9) = 0$

Step 7

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

$4 r - 1 = 0$

$4 r + 9 = 0$

Step 8

Set $4 r - 1$ equal to $0$ and solve for $r$ .

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Step 8.1

Set $4 r - 1$ equal to $0$ .

$4 r - 1 = 0$

Step 8.2

Solve $4 r - 1 = 0$ for $r$ .

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Step 8.2.1

Add $1$ to both sides of the equation.

$4 r = 1$

Step 8.2.2

Divide each term in $4 r = 1$ by $4$ and simplify.

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Step 8.2.2.1

Divide each term in $4 r = 1$ by $4$ .

$\frac{4 r}{4} = \frac{1}{4}$

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $4$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

$\frac{4 r}{4} = \frac{1}{4}$

Step 8.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{1}{4}$

Step 9

Set $4 r + 9$ equal to $0$ and solve for $r$ .

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Step 9.1

Set $4 r + 9$ equal to $0$ .

$4 r + 9 = 0$

Step 9.2

Solve $4 r + 9 = 0$ for $r$ .

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Step 9.2.1

Subtract $9$ from both sides of the equation.

$4 r = - 9$

Step 9.2.2

Divide each term in $4 r = - 9$ by $4$ and simplify.

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Step 9.2.2.1

Divide each term in $4 r = - 9$ by $4$ .

$\frac{4 r}{4} = \frac{- 9}{4}$

Step 9.2.2.2

Simplify the left side.

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Step 9.2.2.2.1

Cancel the common factor of $4$ .

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Step 9.2.2.2.1.1

Cancel the common factor.

$\frac{4 r}{4} = \frac{- 9}{4}$

Step 9.2.2.2.1.2

Divide $r$ by $1$ .

$r = \frac{- 9}{4}$

Step 9.2.2.3

Simplify the right side.

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Step 9.2.2.3.1

Move the negative in front of the fraction.

$r = - \frac{9}{4}$

Step 10

The final solution is all the values that make $1000 (4 r - 1) (4 r + 9) = 0$ true.

$r = \frac{1}{4}, - \frac{9}{4}$