Precalculus Examples

$0.25 (t) = \frac{t}{3 t^{2} + 1}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$1, 3 t^{2} + 1$

Step 1.2

Remove parentheses.

$1, 3 t^{2} + 1$

Step 1.3

The LCM of one and any expression is the expression.

$3 t^{2} + 1$

Step 2

Multiply each term in $0.25 t = \frac{t}{3 t^{2} + 1}$ by $3 t^{2} + 1$ to eliminate the fractions.

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Step 2.1

Multiply each term in $0.25 t = \frac{t}{3 t^{2} + 1}$ by $3 t^{2} + 1$ .

$0.25 t (3 t^{2} + 1) = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify by multiplying through.

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Step 2.2.1.1

Apply the distributive property.

$0.25 t (3 t^{2}) + 0.25 t \cdot 1 = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.1.2

Simplify the expression.

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Step 2.2.1.2.1

Rewrite using the commutative property of multiplication.

$0.25 \cdot 3 t \cdot t^{2} + 0.25 t \cdot 1 = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.1.2.2

Multiply $0.25$ by $1$ .

$0.25 \cdot 3 t \cdot t^{2} + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Multiply $t$ by $t^{2}$ by adding the exponents.

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Step 2.2.2.1.1

Move $t^{2}$ .

$0.25 \cdot 3 (t^{2} t) + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.2.1.2

Multiply $t^{2}$ by $t$ .

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Step 2.2.2.1.2.1

Raise $t$ to the power of $1$ .

$0.25 \cdot 3 (t^{2} t^{1}) + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.2.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$0.25 \cdot 3 t^{2 + 1} + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.2.1.3

Add $2$ and $1$ .

$0.25 \cdot 3 t^{3} + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.2.2.2

Multiply $0.25$ by $3$ .

$0.75 t^{3} + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $3 t^{2} + 1$ .

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Step 2.3.1.1

Cancel the common factor.

$0.75 t^{3} + 0.25 t = \frac{t}{3 t^{2} + 1} (3 t^{2} + 1)$

Step 2.3.1.2

Rewrite the expression.

$0.75 t^{3} + 0.25 t = t$

Step 3

Solve the equation.

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Step 3.1

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

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Step 3.1.1

Subtract $t$ from both sides of the equation.

$0.75 t^{3} + 0.25 t - t = 0$

Step 3.1.2

Subtract $t$ from $0.25 t$ .

$0.75 t^{3} - 0.75 t = 0$

Step 3.2

Factor the left side of the equation.

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Step 3.2.1

Factor $0.75 t$ out of $0.75 t^{3} - 0.75 t$ .

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Step 3.2.1.1

Factor $0.75 t$ out of $0.75 t^{3}$ .

$0.75 t (t^{2}) - 0.75 t = 0$

Step 3.2.1.2

Factor $0.75 t$ out of $- 0.75 t$ .

$0.75 t (t^{2}) + 0.75 t (- 1) = 0$

Step 3.2.1.3

Factor $0.75 t$ out of $0.75 t (t^{2}) + 0.75 t (- 1)$ .

$0.75 t (t^{2} - 1) = 0$

Step 3.2.2

Rewrite $1$ as $1^{2}$ .

$0.75 t (t^{2} - 1^{2}) = 0$

Step 3.2.3

Factor.

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Step 3.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 1$ .

$0.75 t ((t + 1) (t - 1)) = 0$

Step 3.2.3.2

Remove unnecessary parentheses.

$0.75 t (t + 1) (t - 1) = 0$

Step 3.3

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

$t = 0$

$t + 1 = 0$

$t - 1 = 0$

Step 3.4

Set $t$ equal to $0$ .

$t = 0$

Step 3.5

Set $t + 1$ equal to $0$ and solve for $t$ .

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Step 3.5.1

Set $t + 1$ equal to $0$ .

$t + 1 = 0$

Step 3.5.2

Subtract $1$ from both sides of the equation.

$t = - 1$

Step 3.6

Set $t - 1$ equal to $0$ and solve for $t$ .

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Step 3.6.1

Set $t - 1$ equal to $0$ .

$t - 1 = 0$

Step 3.6.2

Add $1$ to both sides of the equation.

$t = 1$

Step 3.7

The final solution is all the values that make $0.75 t (t + 1) (t - 1) = 0$ true.

$t = 0, - 1, 1$