Pre-Algebra Examples

$s = \frac{H}{m t (1 - m t \cdot 2)}$

Step 1

Rewrite the equation as $\frac{H}{m t (1 - m t \cdot 2)} = s$ .

$\frac{H}{m t (1 - m t \cdot 2)} = s$

Step 2

Multiply $2$ by $- 1$ .

$\frac{H}{m t (1 - 2 m t)} = s$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$m t (1 - 2 m t), 1$

Step 3.2

The LCM of one and any expression is the expression.

$m t (1 - 2 m t)$

Step 4

Multiply each term in $\frac{H}{m t (1 - 2 m t)} = s$ by $m t (1 - 2 m t)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{H}{m t (1 - 2 m t)} = s$ by $m t (1 - 2 m t)$ .

$\frac{H}{m t (1 - 2 m t)} (m t (1 - 2 m t)) = s (m t (1 - 2 m t))$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $m t (1 - 2 m t)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{H}{m t (1 - 2 m t)} (m t (1 - 2 m t)) = s (m t (1 - 2 m t))$

Step 4.2.1.2

Rewrite the expression.

$H = s (m t (1 - 2 m t))$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify by multiplying through.

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Step 4.3.1.1

Apply the distributive property.

$H = s (m t \cdot 1 + m t (- 2 m t))$

Step 4.3.1.2

Simplify the expression.

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Step 4.3.1.2.1

Multiply $m$ by $1$ .

$H = s (m t + m t (- 2 m t))$

Step 4.3.1.2.2

Rewrite using the commutative property of multiplication.

$H = s (m t - 2 m t (m t))$

Step 4.3.2

Simplify each term.

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Step 4.3.2.1

Multiply $m$ by $m$ by adding the exponents.

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Step 4.3.2.1.1

Move $m$ .

$H = s (m t - 2 (m \cdot m) t \cdot t)$

Step 4.3.2.1.2

Multiply $m$ by $m$ .

$H = s (m t - 2 m^{2} t \cdot t)$

Step 4.3.2.2

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

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Step 4.3.2.2.1

Move $t$ .

$H = s (m t - 2 m^{2} (t \cdot t))$

Step 4.3.2.2.2

Multiply $t$ by $t$ .

$H = s (m t - 2 m^{2} t^{2})$

Step 4.3.3

Simplify by multiplying through.

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Step 4.3.3.1

Apply the distributive property.

$H = s (m t) + s (- 2 m^{2} t^{2})$

Step 4.3.3.2

Rewrite using the commutative property of multiplication.

$H = s m t - 2 s m^{2} t^{2}$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $s m t - 2 s m^{2} t^{2} = H$ .

$s m t - 2 s m^{2} t^{2} = H$

Step 5.2

Subtract $H$ from both sides of the equation.

$s m t - 2 s m^{2} t^{2} - H = 0$

Step 5.3

Use the quadratic formula to find the solutions.

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

Step 5.4

Substitute the values $a = - 2 s m^{2}$ , $b = s m$ , and $c = - H$ into the quadratic formula and solve for $t$ .

$\frac{- s m \pm \sqrt{{(s m)}^{2} - 4 \cdot (- 2 s m^{2} \cdot (- H))}}{2 (- 2 s m^{2})}$

Step 5.5

Simplify.

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Step 5.5.1

Simplify the numerator.

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Step 5.5.1.1

Apply the product rule to $s m$ .

$t = \frac{- s m \pm \sqrt{s^{2} m^{2} - 4 \cdot (- 2 s m^{2}) \cdot (- 1 H)}}{2 (- 2 s m^{2})}$

Step 5.5.1.2

Multiply $- 4$ by $- 2$ .

$t = \frac{- s m \pm \sqrt{s^{2} m^{2} + 8 s m^{2} \cdot (- 1 H)}}{2 (- 2 s m^{2})}$

Step 5.5.1.3

Multiply $- 1$ by $8$ .

$t = \frac{- s m \pm \sqrt{s^{2} m^{2} - 8 s m^{2} H}}{2 (- 2 s m^{2})}$

Step 5.5.1.4

Factor $s m^{2}$ out of $s^{2} m^{2} - 8 s m^{2} H$ .

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Step 5.5.1.4.1

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

$t = \frac{- s m \pm \sqrt{s m^{2} (s) - 8 s m^{2} H}}{2 (- 2 s m^{2})}$

Step 5.5.1.4.2

Factor $s m^{2}$ out of $- 8 s m^{2} H$ .

$t = \frac{- s m \pm \sqrt{s m^{2} (s) + s m^{2} (- 8 H)}}{2 (- 2 s m^{2})}$

Step 5.5.1.4.3

Factor $s m^{2}$ out of $s m^{2} (s) + s m^{2} (- 8 H)$ .

$t = \frac{- s m \pm \sqrt{s m^{2} (s - 8 H)}}{2 (- 2 s m^{2})}$

Step 5.5.1.5

Rewrite $s m^{2} (s - 8 H)$ as $m^{2} (s (s - 8 H))$ .

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Step 5.5.1.5.1

Reorder $s$ and $m^{2}$ .

$t = \frac{- s m \pm \sqrt{m^{2} s (s - 8 H)}}{2 (- 2 s m^{2})}$

Step 5.5.1.5.2

Add parentheses.

$t = \frac{- s m \pm \sqrt{m^{2} (s (s - 8 H))}}{2 (- 2 s m^{2})}$

Step 5.5.1.6

Pull terms out from under the radical.

$t = \frac{- s m \pm m \sqrt{s (s - 8 H)}}{2 (- 2 s m^{2})}$

Step 5.5.2

Multiply $- 2$ by $2$ .

$t = \frac{- s m \pm m \sqrt{s (s - 8 H)}}{- 4 (s m^{2})}$

Step 5.5.3

Simplify $\frac{- s m \pm m \sqrt{s (s - 8 H)}}{- 4 s m^{2}}$ .

$t = \frac{s \pm \sqrt{s (s - 8 H)}}{4 s m}$

Step 5.6

The final answer is the combination of both solutions.

$t = \frac{s + \sqrt{s (s - 8 H)}}{4 s m}$

$t = \frac{s - \sqrt{s (s - 8 H)}}{4 s m}$

$t = \frac{s + \sqrt{s (s - 8 H)}}{4 s m}$

$t = \frac{s - \sqrt{s (s - 8 H)}}{4 s m}$