Precalculus Examples

$0 = 32 t - 16 t^{2}$

Step 1

Rewrite the equation as $32 t - 16 t^{2} = 0$ .

$32 t - 16 t^{2} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

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

$32 u - 16 u^{2} = 0$

Step 2.2

Factor $16 u$ out of $32 u - 16 u^{2}$ .

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Step 2.2.1

Factor $16 u$ out of $32 u$ .

$16 u (2) - 16 u^{2} = 0$

Step 2.2.2

Factor $16 u$ out of $- 16 u^{2}$ .

$16 u (2) + 16 u (- u) = 0$

Step 2.2.3

Factor $16 u$ out of $16 u (2) + 16 u (- u)$ .

$16 u (2 - u) = 0$

Step 2.3

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

$16 t (2 - t) = 0$

Step 3

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

$t = 0$

$2 - t = 0$

Step 4

Set $t$ equal to $0$ .

$t = 0$

Step 5

Set $2 - t$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $2 - t$ equal to $0$ .

$2 - t = 0$

Step 5.2

Solve $2 - t = 0$ for $t$ .

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Step 5.2.1

Subtract $2$ from both sides of the equation.

$- t = - 2$

Step 5.2.2

Divide each term in $- t = - 2$ by $- 1$ and simplify.

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Step 5.2.2.1

Divide each term in $- t = - 2$ by $- 1$ .

$\frac{- t}{- 1} = \frac{- 2}{- 1}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{t}{1} = \frac{- 2}{- 1}$

Step 5.2.2.2.2

Divide $t$ by $1$ .

$t = \frac{- 2}{- 1}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Divide $- 2$ by $- 1$ .

$t = 2$

Step 6

The final solution is all the values that make $16 t (2 - t) = 0$ true.

$t = 0, 2$