Precalculus Examples

$s (t) = \frac{- 7 t}{(t - 4) (t - 8)}$

Step 1

Move the negative in front of the fraction.

$s t = - \frac{7 t}{(t - 4) (t - 8)}$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, (t - 4) (t - 8)$

Step 2.2

The LCM of one and any expression is the expression.

$(t - 4) (t - 8)$

Step 3

Multiply each term in $s t = - \frac{7 t}{(t - 4) (t - 8)}$ by $(t - 4) (t - 8)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $s t = - \frac{7 t}{(t - 4) (t - 8)}$ by $(t - 4) (t - 8)$ .

$s t ((t - 4) (t - 8)) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Expand $(t - 4) (t - 8)$ using the FOIL Method.

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$s t (t (t - 8) - 4 (t - 8)) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.1.2

Apply the distributive property.

$s t (t \cdot t + t \cdot - 8 - 4 (t - 8)) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.1.3

Apply the distributive property.

$s t (t \cdot t + t \cdot - 8 - 4 t - 4 \cdot - 8) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.2

Simplify and combine like terms.

Tap for more steps...

Step 3.2.2.1

Simplify each term.

Tap for more steps...

Step 3.2.2.1.1

Multiply $t$ by $t$ .

$s t (t^{2} + t \cdot - 8 - 4 t - 4 \cdot - 8) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.2.1.2

Move $- 8$ to the left of $t$ .

$s t (t^{2} - 8 \cdot t - 4 t - 4 \cdot - 8) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.2.1.3

Multiply $- 4$ by $- 8$ .

$s t (t^{2} - 8 t - 4 t + 32) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.2.2

Subtract $4 t$ from $- 8 t$ .

$s t (t^{2} - 12 t + 32) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.3

Apply the distributive property.

$s t \cdot t^{2} + s t (- 12 t) + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4

Simplify.

Tap for more steps...

Step 3.2.4.1

Multiply $t$ by $t^{2}$ by adding the exponents.

Tap for more steps...

Step 3.2.4.1.1

Move $t^{2}$ .

$s (t^{2} t) + s t (- 12 t) + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4.1.2

Multiply $t^{2}$ by $t$ .

Tap for more steps...

Step 3.2.4.1.2.1

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

$s (t^{2} t^{1}) + s t (- 12 t) + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4.1.2.2

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

$s t^{2 + 1} + s t (- 12 t) + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4.1.3

Add $2$ and $1$ .

$s t^{3} + s t (- 12 t) + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4.2

Rewrite using the commutative property of multiplication.

$s t^{3} - 12 s t \cdot t + s t \cdot 32 = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.4.3

Move $32$ to the left of $s t$ .

$s t^{3} - 12 s t \cdot t + 32 \cdot (s t) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.5

Multiply $t$ by $t$ by adding the exponents.

Tap for more steps...

Step 3.2.5.1

Move $t$ .

$s t^{3} - 12 s (t \cdot t) + 32 \cdot (s t) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.2.5.2

Multiply $t$ by $t$ .

$s t^{3} - 12 s t^{2} + 32 \cdot (s t) = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

$s t^{3} - 12 s t^{2} + 32 s t = - \frac{7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Cancel the common factor of $(t - 4) (t - 8)$ .

Tap for more steps...

Step 3.3.1.1

Move the leading negative in $- \frac{7 t}{(t - 4) (t - 8)}$ into the numerator.

$s t^{3} - 12 s t^{2} + 32 s t = \frac{- 7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.3.1.2

Cancel the common factor.

$s t^{3} - 12 s t^{2} + 32 s t = \frac{- 7 t}{(t - 4) (t - 8)} ((t - 4) (t - 8))$

Step 3.3.1.3

Rewrite the expression.

$s t^{3} - 12 s t^{2} + 32 s t = - 7 t$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Add $7 t$ to both sides of the equation.

$s t^{3} - 12 s t^{2} + 32 s t + 7 t = 0$

Step 4.2

Factor $t$ out of $s t^{3} - 12 s t^{2} + 32 s t + 7 t$ .

Tap for more steps...

Step 4.2.1

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

$t (s t^{2}) - 12 s t^{2} + 32 s t + 7 t = 0$

Step 4.2.2

Factor $t$ out of $- 12 s t^{2}$ .

$t (s t^{2}) + t (- 12 s t) + 32 s t + 7 t = 0$

Step 4.2.3

Factor $t$ out of $32 s t$ .

$t (s t^{2}) + t (- 12 s t) + t (32 s) + 7 t = 0$

Step 4.2.4

Factor $t$ out of $7 t$ .

$t (s t^{2}) + t (- 12 s t) + t (32 s) + t \cdot 7 = 0$

Step 4.2.5

Factor $t$ out of $t (s t^{2}) + t (- 12 s t)$ .

$t (s t^{2} - 12 s t) + t (32 s) + t \cdot 7 = 0$

Step 4.2.6

Factor $t$ out of $t (s t^{2} - 12 s t) + t (32 s)$ .

$t (s t^{2} - 12 s t + 32 s) + t \cdot 7 = 0$

Step 4.2.7

Factor $t$ out of $t (s t^{2} - 12 s t + 32 s) + t \cdot 7$ .

$t (s t^{2} - 12 s t + 32 s + 7) = 0$

Step 4.3

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

$t = 0$

$s t^{2} - 12 s t + 32 s + 7 = 0$

Step 4.4

Set $t$ equal to $0$ .

$t = 0$

Step 4.5

Set $s t^{2} - 12 s t + 32 s + 7$ equal to $0$ and solve for $t$ .

Tap for more steps...

Step 4.5.1

Set $s t^{2} - 12 s t + 32 s + 7$ equal to $0$ .

$s t^{2} - 12 s t + 32 s + 7 = 0$

Step 4.5.2

Solve $s t^{2} - 12 s t + 32 s + 7 = 0$ for $t$ .

Tap for more steps...

Step 4.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.5.2.2

Substitute the values $a = s$ , $b = - 12 s$ , and $c = 32 s + 7$ into the quadratic formula and solve for $t$ .

$\frac{12 s \pm \sqrt{{(- 12 s)}^{2} - 4 \cdot (s \cdot (32 s + 7))}}{2 s}$

Step 4.5.2.3

Simplify.

Tap for more steps...

Step 4.5.2.3.1

Simplify the numerator.

Tap for more steps...

Step 4.5.2.3.1.1

Add parentheses.

$t = \frac{12 s \pm \sqrt{{(- 12 s)}^{2} - 4 \cdot (s \cdot (32 s + 7))}}{2 s}$

Step 4.5.2.3.1.2

Let $u = s \cdot (32 s + 7)$ . Substitute $u$ for all occurrences of $s \cdot (32 s + 7)$ .

Tap for more steps...

Step 4.5.2.3.1.2.1

Apply the product rule to $- 12 s$ .

$t = \frac{12 s \pm \sqrt{{(- 12)}^{2} s^{2} - 4 \cdot u}}{2 s}$

Step 4.5.2.3.1.2.2

Raise $- 12$ to the power of $2$ .

$t = \frac{12 s \pm \sqrt{144 s^{2} - 4 u}}{2 s}$

Step 4.5.2.3.1.3

Factor $4$ out of $144 s^{2} - 4 u$ .

Tap for more steps...

Step 4.5.2.3.1.3.1

Factor $4$ out of $144 s^{2}$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2}) - 4 u}}{2 s}$

Step 4.5.2.3.1.3.2

Factor $4$ out of $- 4 u$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2}) + 4 (- u)}}{2 s}$

Step 4.5.2.3.1.3.3

Factor $4$ out of $4 (36 s^{2}) + 4 (- u)$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - u)}}{2 s}$

Step 4.5.2.3.1.4

Replace all occurrences of $u$ with $s \cdot (32 s + 7)$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (s \cdot (32 s + 7)))}}{2 s}$

Step 4.5.2.3.1.5

Simplify.

Tap for more steps...

Step 4.5.2.3.1.5.1

Simplify each term.

Tap for more steps...

Step 4.5.2.3.1.5.1.1

Apply the distributive property.

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (s (32 s) + s \cdot 7))}}{2 s}$

Step 4.5.2.3.1.5.1.2

Rewrite using the commutative property of multiplication.

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 s \cdot s + s \cdot 7))}}{2 s}$

Step 4.5.2.3.1.5.1.3

Move $7$ to the left of $s$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 s \cdot s + 7 \cdot s))}}{2 s}$

Step 4.5.2.3.1.5.1.4

Multiply $s$ by $s$ by adding the exponents.

Tap for more steps...

Step 4.5.2.3.1.5.1.4.1

Move $s$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 (s \cdot s) + 7 \cdot s))}}{2 s}$

Step 4.5.2.3.1.5.1.4.2

Multiply $s$ by $s$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 s^{2} + 7 \cdot s))}}{2 s}$

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 s^{2} + 7 s))}}{2 s}$

Step 4.5.2.3.1.5.1.5

Apply the distributive property.

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - (32 s^{2}) - (7 s))}}{2 s}$

Step 4.5.2.3.1.5.1.6

Multiply $32$ by $- 1$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - 32 s^{2} - (7 s))}}{2 s}$

Step 4.5.2.3.1.5.1.7

Multiply $7$ by $- 1$ .

$t = \frac{12 s \pm \sqrt{4 (36 s^{2} - 32 s^{2} - 7 s)}}{2 s}$

Step 4.5.2.3.1.5.2

Subtract $32 s^{2}$ from $36 s^{2}$ .

$t = \frac{12 s \pm \sqrt{4 (4 s^{2} - 7 s)}}{2 s}$

Step 4.5.2.3.1.6

Factor $s$ out of $4 s^{2} - 7 s$ .

Tap for more steps...

Step 4.5.2.3.1.6.1

Factor $s$ out of $4 s^{2}$ .

$t = \frac{12 s \pm \sqrt{4 (s (4 s) - 7 s)}}{2 s}$

Step 4.5.2.3.1.6.2

Factor $s$ out of $- 7 s$ .

$t = \frac{12 s \pm \sqrt{4 (s (4 s) + s \cdot - 7)}}{2 s}$

Step 4.5.2.3.1.6.3

Factor $s$ out of $s (4 s) + s \cdot - 7$ .

$t = \frac{12 s \pm \sqrt{4 (s (4 s - 7))}}{2 s}$

$t = \frac{12 s \pm \sqrt{4 s (4 s - 7)}}{2 s}$

Step 4.5.2.3.1.7

Rewrite $4 s (4 s - 7)$ as $2^{2} (s (2^{2} s - 7))$ .

Tap for more steps...

Step 4.5.2.3.1.7.1

Rewrite $4$ as $2^{2}$ .

$t = \frac{12 s \pm \sqrt{2^{2} s (4 s - 7)}}{2 s}$

Step 4.5.2.3.1.7.2

Rewrite $4$ as $2^{2}$ .

$t = \frac{12 s \pm \sqrt{2^{2} s (2^{2} s - 7)}}{2 s}$

Step 4.5.2.3.1.7.3

Add parentheses.

$t = \frac{12 s \pm \sqrt{2^{2} (s (2^{2} s - 7))}}{2 s}$

Step 4.5.2.3.1.8

Pull terms out from under the radical.

$t = \frac{12 s \pm 2 \sqrt{s (2^{2} s - 7)}}{2 s}$

Step 4.5.2.3.1.9

Raise $2$ to the power of $2$ .

$t = \frac{12 s \pm 2 \sqrt{s (4 s - 7)}}{2 s}$

Step 4.5.2.3.2

Simplify $\frac{12 s \pm 2 \sqrt{s (4 s - 7)}}{2 s}$ .

$t = \frac{6 s \pm \sqrt{s (4 s - 7)}}{s}$

Step 4.5.2.4

The final answer is the combination of both solutions.

$t = \frac{6 s + \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s - \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s + \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s - \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s + \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s - \sqrt{s (4 s - 7)}}{s}$

Step 4.6

The final solution is all the values that make $t (s t^{2} - 12 s t + 32 s + 7) = 0$ true.

$t = 0$

$t = \frac{6 s + \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s - \sqrt{s (4 s - 7)}}{s}$

$t = 0$

$t = \frac{6 s + \sqrt{s (4 s - 7)}}{s}$

$t = \frac{6 s - \sqrt{s (4 s - 7)}}{s}$