Solve by Factoring 1-4/t-21/(t^2)=0

$1 - \frac{4}{t} - \frac{21}{t^{2}} = 0$

Step 1

Reorder terms.

$- \frac{4}{t} - \frac{21}{t^{2}} + 1 = 0$

Step 2

To write $- \frac{4}{t}$ as a fraction with a common denominator, multiply by $\frac{t}{t}$ .

$- \frac{4}{t} \cdot \frac{t}{t} - \frac{21}{t^{2}} + 1 = 0$

Step 3

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

Multiply $\frac{4}{t}$ by $\frac{t}{t}$ .

$- \frac{4 t}{t \cdot t} - \frac{21}{t^{2}} + 1 = 0$

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

$- \frac{4 t}{t \cdot t} - \frac{21}{t^{2}} + 1 = 0$

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

$- \frac{4 t}{t \cdot t} - \frac{21}{t^{2}} + 1 = 0$

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

$- \frac{4 t}{t^{1 + 1}} - \frac{21}{t^{2}} + 1 = 0$

Add $1$ and $1$ .

$- \frac{4 t}{t^{2}} - \frac{21}{t^{2}} + 1 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{- 4 t - 21}{t^{2}} + 1 = 0$

Step 5

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

$\frac{- 4 t - 21}{t^{2}} + \frac{t^{2}}{t^{2}} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{- 4 t - 21 + t^{2}}{t^{2}} = 0$

Step 7

Factor $- 4 t - 21 + t^{2}$ 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 $- 21$ and whose sum is $- 4$ .

$- 7, 3$

Write the factored form using these integers.

$\frac{(t - 7) (t + 3)}{t^{2}} = 0$

Step 8

Set the numerator equal to zero.

$(t - 7) (t + 3) = 0$

Step 9

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 $0$ .

$t - 7 = 0$

$t + 3 = 0$

Set $t - 7$ equal to $0$ and solve for $t$ .

Set $t - 7$ equal to $0$ .

$t - 7 = 0$

Add $7$ to both sides of the equation.

$t = 7$

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

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

$t + 3 = 0$

Subtract $3$ from both sides of the equation.

$t = - 3$

The final solution is all the values that make $(t - 7) (t + 3) = 0$ true.

$t = 7, - 3$

Solve by Factoring 1-4/t-21/(t^2)=0

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