Precalculus Examples

$h = b t + \frac{1}{2} \cdot (a t^{2})$

Step 1

Rewrite the equation as $b t + \frac{1}{2} \cdot (a t^{2}) = h$ .

$b t + \frac{1}{2} \cdot (a t^{2}) = h$

Step 2

Multiply $\frac{1}{2} (a t^{2})$ .

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Step 2.1

Combine $a$ and $\frac{1}{2}$ .

$b t + \frac{a}{2} t^{2} = h$

Step 2.2

Combine $\frac{a}{2}$ and $t^{2}$ .

$b t + \frac{a t^{2}}{2} = h$

Step 3

Subtract $b t$ from both sides of the equation.

$\frac{a t^{2}}{2} = h - b t$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{a t^{2}}{2} = 2 (h - b t)$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

$2 \frac{a t^{2}}{2} = 2 (h - b t)$

Step 5.1.1.2

Rewrite the expression.

$a t^{2} = 2 (h - b t)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (h - b t)$ .

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Step 5.2.1.1

Apply the distributive property.

$a t^{2} = 2 h + 2 (- b t)$

Step 5.2.1.2

Multiply $- 1$ by $2$ .

$a t^{2} = 2 h - 2 b t$

Step 6

Divide each term in $a t^{2} = 2 h - 2 b t$ by $t^{2}$ and simplify.

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Step 6.1

Divide each term in $a t^{2} = 2 h - 2 b t$ by $t^{2}$ .

$\frac{a t^{2}}{t^{2}} = \frac{2 h}{t^{2}} + \frac{- 2 b t}{t^{2}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{a t^{2}}{t^{2}} = \frac{2 h}{t^{2}} + \frac{- 2 b t}{t^{2}}$

Step 6.2.1.2

Divide $a$ by $1$ .

$a = \frac{2 h}{t^{2}} + \frac{- 2 b t}{t^{2}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Cancel the common factor of $t$ and $t^{2}$ .

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Step 6.3.1.1.1

Factor $t$ out of $- 2 b t$ .

$a = \frac{2 h}{t^{2}} + \frac{t (- 2 b)}{t^{2}}$

Step 6.3.1.1.2

Cancel the common factors.

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Step 6.3.1.1.2.1

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

$a = \frac{2 h}{t^{2}} + \frac{t (- 2 b)}{t \cdot t}$

Step 6.3.1.1.2.2

Cancel the common factor.

$a = \frac{2 h}{t^{2}} + \frac{t (- 2 b)}{t \cdot t}$

Step 6.3.1.1.2.3

Rewrite the expression.

$a = \frac{2 h}{t^{2}} + \frac{- 2 b}{t}$

Step 6.3.1.2

Move the negative in front of the fraction.

$a = \frac{2 h}{t^{2}} - \frac{2 b}{t}$