Precalculus Examples

$9 t^{3} - 25 t = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $t$ out of $9 t^{3} - 25 t$ .

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Step 1.1.1

Factor $t$ out of $9 t^{3}$ .

$t (9 t^{2}) - 25 t = 0$

Step 1.1.2

Factor $t$ out of $- 25 t$ .

$t (9 t^{2}) + t \cdot - 25 = 0$

Step 1.1.3

Factor $t$ out of $t (9 t^{2}) + t \cdot - 25$ .

$t (9 t^{2} - 25) = 0$

Step 1.2

Rewrite $9 t^{2}$ as ${(3 t)}^{2}$ .

$t ({(3 t)}^{2} - 25) = 0$

Step 1.3

Rewrite $25$ as $5^{2}$ .

$t ({(3 t)}^{2} - 5^{2}) = 0$

Step 1.4

Factor.

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Step 1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 t$ and $b = 5$ .

$t ((3 t + 5) (3 t - 5)) = 0$

Step 1.4.2

Remove unnecessary parentheses.

$t (3 t + 5) (3 t - 5) = 0$

Step 2

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

$t = 0$

$3 t + 5 = 0$

$3 t - 5 = 0$

Step 3

Set $t$ equal to $0$ .

$t = 0$

Step 4

Set $3 t + 5$ equal to $0$ and solve for $t$ .

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Step 4.1

Set $3 t + 5$ equal to $0$ .

$3 t + 5 = 0$

Step 4.2

Solve $3 t + 5 = 0$ for $t$ .

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Step 4.2.1

Subtract $5$ from both sides of the equation.

$3 t = - 5$

Step 4.2.2

Divide each term in $3 t = - 5$ by $3$ and simplify.

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Step 4.2.2.1

Divide each term in $3 t = - 5$ by $3$ .

$\frac{3 t}{3} = \frac{- 5}{3}$

Step 4.2.2.2

Simplify the left side.

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Step 4.2.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{- 5}{3}$

Step 4.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 5}{3}$

Step 4.2.2.3

Simplify the right side.

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Step 4.2.2.3.1

Move the negative in front of the fraction.

$t = - \frac{5}{3}$

Step 5

Set $3 t - 5$ equal to $0$ and solve for $t$ .

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Step 5.1

Set $3 t - 5$ equal to $0$ .

$3 t - 5 = 0$

Step 5.2

Solve $3 t - 5 = 0$ for $t$ .

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Step 5.2.1

Add $5$ to both sides of the equation.

$3 t = 5$

Step 5.2.2

Divide each term in $3 t = 5$ by $3$ and simplify.

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Step 5.2.2.1

Divide each term in $3 t = 5$ by $3$ .

$\frac{3 t}{3} = \frac{5}{3}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{3 t}{3} = \frac{5}{3}$

Step 5.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{5}{3}$

Step 6

The final solution is all the values that make $t (3 t + 5) (3 t - 5) = 0$ true.

$t = 0, - \frac{5}{3}, \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$t = 0, - \frac{5}{3}, \frac{5}{3}$

Decimal Form:

$t = 0, - 1.\overline{6}, 1.\overline{6}$

Mixed Number Form:

$t = 0, - 1 \frac{2}{3}, 1 \frac{2}{3}$