Calculus Examples

$2500 = 100 (1 + 0.2 t + 0.02 t^{2})$

Step 1

Rewrite the equation as $100 (1 + 0.2 t + 0.02 t^{2}) = 2500$ .

$100 (1 + 0.2 t + 0.02 t^{2}) = 2500$

Step 2

Divide each term in $100 (1 + 0.2 t + 0.02 t^{2}) = 2500$ by $100$ and simplify.

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Step 2.1

Divide each term in $100 (1 + 0.2 t + 0.02 t^{2}) = 2500$ by $100$ .

$\frac{100 (1 + 0.2 t + 0.02 t^{2})}{100} = \frac{2500}{100}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $100$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{100 (1 + 0.2 t + 0.02 t^{2})}{100} = \frac{2500}{100}$

Step 2.2.1.2

Divide $1 + 0.2 t + 0.02 t^{2}$ by $1$ .

$1 + 0.2 t + 0.02 t^{2} = \frac{2500}{100}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $2500$ by $100$ .

$1 + 0.2 t + 0.02 t^{2} = 25$

Step 3

Subtract $25$ from both sides of the equation.

$1 + 0.2 t + 0.02 t^{2} - 25 = 0$

Step 4

Subtract $25$ from $1$ .

$0.2 t + 0.02 t^{2} - 24 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $0.02$ out of $0.2 t + 0.02 t^{2} - 24$ .

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Step 5.1.1

Factor $0.02$ out of $0.2 t$ .

$0.02 (10 t) + 0.02 t^{2} - 24 = 0$

Step 5.1.2

Factor $0.02$ out of $0.02 t^{2}$ .

$0.02 (10 t) + 0.02 (t^{2}) - 24 = 0$

Step 5.1.3

Factor $0.02$ out of $- 24$ .

$0.02 (10 t) + 0.02 t^{2} + 0.02 \cdot - 1200 = 0$

Step 5.1.4

Factor $0.02$ out of $0.02 (10 t) + 0.02 t^{2}$ .

$0.02 (10 t + t^{2}) + 0.02 \cdot - 1200 = 0$

Step 5.1.5

Factor $0.02$ out of $0.02 (10 t + t^{2}) + 0.02 \cdot - 1200$ .

$0.02 (10 t + t^{2} - 1200) = 0$

Step 5.2

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

$0.02 (10 u + u^{2} - 1200) = 0$

Step 5.3

Factor $10 u + u^{2} - 1200$ using the AC method.

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Step 5.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 1200$ and whose sum is $10$ .

$- 30, 40$

Step 5.3.2

Write the factored form using these integers.

$0.02 ((u - 30) (u + 40)) = 0$

Step 5.4

Factor.

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Step 5.4.1

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

$0.02 ((t - 30) (t + 40)) = 0$

Step 5.4.2

Remove unnecessary parentheses.

$0.02 (t - 30) (t + 40) = 0$

Step 6

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

$t - 30 = 0$

$t + 40 = 0$

Step 7

Set $t - 30$ equal to $0$ and solve for $t$ .

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Step 7.1

Set $t - 30$ equal to $0$ .

$t - 30 = 0$

Step 7.2

Add $30$ to both sides of the equation.

$t = 30$

Step 8

Set $t + 40$ equal to $0$ and solve for $t$ .

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Step 8.1

Set $t + 40$ equal to $0$ .

$t + 40 = 0$

Step 8.2

Subtract $40$ from both sides of the equation.

$t = - 40$

Step 9

The final solution is all the values that make $0.02 (t - 30) (t + 40) = 0$ true.

$t = 30, - 40$