Algebra Examples

$9 - \frac{1}{t} - \frac{8}{t^{2}} = 0$

Step 1

Reorder terms.

$- \frac{1}{t} - \frac{8}{t^{2}} + 9 = 0$

Step 2

To write $- \frac{1}{t}$ as a fraction with a common denominator, multiply by $\frac{t}{t}$ .

$- \frac{1}{t} \cdot \frac{t}{t} - \frac{8}{t^{2}} + 9 = 0$

Step 3

Write each expression with a common denominator of $t^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{t}$ by $\frac{t}{t}$ .

$- \frac{t}{t \cdot t} - \frac{8}{t^{2}} + 9 = 0$

Step 3.2

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

$- \frac{t}{t \cdot t} - \frac{8}{t^{2}} + 9 = 0$

Step 3.3

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

$- \frac{t}{t \cdot t} - \frac{8}{t^{2}} + 9 = 0$

Step 3.4

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

$- \frac{t}{t^{1 + 1}} - \frac{8}{t^{2}} + 9 = 0$

Step 3.5

Add $1$ and $1$ .

$- \frac{t}{t^{2}} - \frac{8}{t^{2}} + 9 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{- t - 8}{t^{2}} + 9 = 0$

Step 5

To write $9$ as a fraction with a common denominator, multiply by $\frac{t^{2}}{t^{2}}$ .

$\frac{- t - 8}{t^{2}} + \frac{9 t^{2}}{t^{2}} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- t - 8 + 9 t^{2}}{t^{2}} = 0$

Step 7

Factor by grouping.

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Step 7.1

Reorder terms.

$\frac{9 t^{2} - t - 8}{t^{2}} = 0$

Step 7.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 9 \cdot - 8 = - 72$ and whose sum is $b = - 1$ .

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Step 7.2.1

Factor $- 1$ out of $- t$ .

$\frac{9 t^{2} - (t) - 8}{t^{2}} = 0$

Step 7.2.2

Rewrite $- 1$ as $8$ plus $- 9$

$\frac{9 t^{2} + (8 - 9) t - 8}{t^{2}} = 0$

Step 7.2.3

Apply the distributive property.

$\frac{9 t^{2} + 8 t - 9 t - 8}{t^{2}} = 0$

Step 7.3

Factor out the greatest common factor from each group.

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Step 7.3.1

Group the first two terms and the last two terms.

$\frac{(9 t^{2} + 8 t) - 9 t - 8}{t^{2}} = 0$

Step 7.3.2

Factor out the greatest common factor (GCF) from each group.

$\frac{t (9 t + 8) - (9 t + 8)}{t^{2}} = 0$

Step 7.4

Factor the polynomial by factoring out the greatest common factor, $9 t + 8$ .

$\frac{(9 t + 8) (t - 1)}{t^{2}} = 0$

Step 8

Set the numerator equal to zero.

$(9 t + 8) (t - 1) = 0$

Step 9

Solve the equation for $t$ .

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Step 9.1

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

$9 t + 8 = 0$

$t - 1 = 0$

Step 9.2

Set $9 t + 8$ equal to $0$ and solve for $t$ .

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Step 9.2.1

Set $9 t + 8$ equal to $0$ .

$9 t + 8 = 0$

Step 9.2.2

Solve $9 t + 8 = 0$ for $t$ .

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Step 9.2.2.1

Subtract $8$ from both sides of the equation.

$9 t = - 8$

Step 9.2.2.2

Divide each term in $9 t = - 8$ by $9$ and simplify.

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Step 9.2.2.2.1

Divide each term in $9 t = - 8$ by $9$ .

$\frac{9 t}{9} = \frac{- 8}{9}$

Step 9.2.2.2.2

Simplify the left side.

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Step 9.2.2.2.2.1

Cancel the common factor of $9$ .

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Step 9.2.2.2.2.1.1

Cancel the common factor.

$\frac{9 t}{9} = \frac{- 8}{9}$

Step 9.2.2.2.2.1.2

Divide $t$ by $1$ .

$t = \frac{- 8}{9}$

Step 9.2.2.2.3

Simplify the right side.

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Step 9.2.2.2.3.1

Move the negative in front of the fraction.

$t = - \frac{8}{9}$

Step 9.3

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

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Step 9.3.1

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

$t - 1 = 0$

Step 9.3.2

Add $1$ to both sides of the equation.

$t = 1$

Step 9.4

The final solution is all the values that make $(9 t + 8) (t - 1) = 0$ true.

$t = - \frac{8}{9}, 1$