Basic Math Examples

$- 64 t^{2} + qt + b = h$

Step 1

Move all terms not containing $t$ to the right side of the equation.

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Step 1.1

Subtract $qt$ from both sides of the equation.

$- 64 t^{2} + b = h - qt$

Step 1.2

Subtract $b$ from both sides of the equation.

$- 64 t^{2} = h - qt - b$

Step 2

Divide each term in $- 64 t^{2} = h - qt - b$ by $- 64$ and simplify.

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Step 2.1

Divide each term in $- 64 t^{2} = h - qt - b$ by $- 64$ .

$\frac{- 64 t^{2}}{- 64} = \frac{h}{- 64} + \frac{- qt}{- 64} + \frac{- b}{- 64}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 64$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 64 t^{2}}{- 64} = \frac{h}{- 64} + \frac{- qt}{- 64} + \frac{- b}{- 64}$

Step 2.2.1.2

Divide $t^{2}$ by $1$ .

$t^{2} = \frac{h}{- 64} + \frac{- qt}{- 64} + \frac{- b}{- 64}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Move the negative in front of the fraction.

$t^{2} = - \frac{h}{64} + \frac{- qt}{- 64} + \frac{- b}{- 64}$

Step 2.3.1.2

Dividing two negative values results in a positive value.

$t^{2} = - \frac{h}{64} + \frac{qt}{64} + \frac{- b}{- 64}$

Step 2.3.1.3

Dividing two negative values results in a positive value.

$t^{2} = - \frac{h}{64} + \frac{qt}{64} + \frac{b}{64}$

Step 3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$t = \pm \sqrt{- \frac{h}{64} + \frac{qt}{64} + \frac{b}{64}}$

Step 4

Simplify $\pm \sqrt{- \frac{h}{64} + \frac{qt}{64} + \frac{b}{64}}$ .

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Step 4.1

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{- h + qt}{64} + \frac{b}{64}}$

Step 4.2

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{- h + qt + b}{64}}$

Step 4.3

Rewrite $\frac{- h + qt + b}{64}$ as ${(\frac{1}{8})}^{2} (- h + qt + b)$ .

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Step 4.3.1

Factor the perfect power $1^{2}$ out of $- h + qt + b$ .

$t = \pm \sqrt{\frac{1^{2} (- h + qt + b)}{64}}$

Step 4.3.2

Factor the perfect power $8^{2}$ out of $64$ .

$t = \pm \sqrt{\frac{1^{2} (- h + qt + b)}{8^{2} \cdot 1}}$

Step 4.3.3

Rearrange the fraction $\frac{1^{2} (- h + qt + b)}{8^{2} \cdot 1}$ .

$t = \pm \sqrt{{(\frac{1}{8})}^{2} (- h + qt + b)}$

Step 4.4

Pull terms out from under the radical.

$t = \pm \frac{1}{8} \sqrt{- h + qt + b}$

Step 4.5

Combine $\frac{1}{8}$ and $\sqrt{- h + qt + b}$ .

$t = \pm \frac{\sqrt{- h + qt + b}}{8}$

Step 5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 5.1

First, use the positive value of the $\pm$ to find the first solution.

$t = \frac{\sqrt{- h + qt + b}}{8}$

Step 5.2

Next, use the negative value of the $\pm$ to find the second solution.

$t = - \frac{\sqrt{- h + qt + b}}{8}$

Step 5.3

The complete solution is the result of both the positive and negative portions of the solution.

$t = \frac{\sqrt{- h + qt + b}}{8}, - \frac{\sqrt{- h + qt + b}}{8}$