Solve by Factoring 1-2/t-80/(t^2)=0

$1 - \frac{2}{t} - \frac{80}{t^{2}} = 0$

Step 1

Reorder terms.

$- \frac{2}{t} - \frac{80}{t^{2}} + 1 = 0$

Step 2

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

$- \frac{2}{t} \cdot \frac{t}{t} - \frac{80}{t^{2}} + 1 = 0$

Step 3

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

Multiply $\frac{2}{t}$ by $\frac{t}{t}$ .

$- \frac{2 t}{t \cdot t} - \frac{80}{t^{2}} + 1 = 0$

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

$- \frac{2 t}{t \cdot t} - \frac{80}{t^{2}} + 1 = 0$

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

$- \frac{2 t}{t \cdot t} - \frac{80}{t^{2}} + 1 = 0$

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

$- \frac{2 t}{t^{1 + 1}} - \frac{80}{t^{2}} + 1 = 0$

Add $1$ and $1$ .

$- \frac{2 t}{t^{2}} - \frac{80}{t^{2}} + 1 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{- 2 t - 80}{t^{2}} + 1 = 0$

Step 5

Factor $2$ out of $- 2 t - 80$ .

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

$\frac{2 (- t) - 80}{t^{2}} + 1 = 0$

Factor $2$ out of $- 80$ .

$\frac{2 (- t) + 2 (- 40)}{t^{2}} + 1 = 0$

Factor $2$ out of $2 (- t) + 2 (- 40)$ .

$\frac{2 (- t - 40)}{t^{2}} + 1 = 0$

Step 6

Write $1$ as a fraction with a common denominator.

$\frac{2 (- t - 40)}{t^{2}} + \frac{t^{2}}{t^{2}} = 0$

Step 7

Combine the numerators over the common denominator.

$\frac{2 (- t - 40) + t^{2}}{t^{2}} = 0$

Step 8

Simplify the numerator.

Apply the distributive property.

$\frac{2 (- t) + 2 \cdot - 40 + t^{2}}{t^{2}} = 0$

Multiply $- 1$ by $2$ .

$\frac{- 2 t + 2 \cdot - 40 + t^{2}}{t^{2}} = 0$

Multiply $2$ by $- 40$ .

$\frac{- 2 t - 80 + t^{2}}{t^{2}} = 0$

Reorder terms.

$\frac{t^{2} - 2 t - 80}{t^{2}} = 0$

Factor $t^{2} - 2 t - 80$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 80$ and whose sum is $- 2$ .

$- 10, 8$

Write the factored form using these integers.

$\frac{(t - 10) (t + 8)}{t^{2}} = 0$

Step 9

Set the numerator equal to zero.

$(t - 10) (t + 8) = 0$

Step 10

Solve the equation for $t$ .

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

$t - 10 = 0$

$t + 8 = 0$

Set $t - 10$ equal to $0$ and solve for $t$ .

Set $t - 10$ equal to $0$ .

$t - 10 = 0$

Add $10$ to both sides of the equation.

$t = 10$

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

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

$t + 8 = 0$

Subtract $8$ from both sides of the equation.

$t = - 8$

The final solution is all the values that make $(t - 10) (t + 8) = 0$ true.

$t = 10, - 8$

Solve by Factoring 1-2/t-80/(t^2)=0

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