Pre-Algebra Examples

$s = - 16 t^{2} + 676$

Step 1

Rewrite the equation as $- 16 t^{2} + 676 = s$ .

$- 16 t^{2} + 676 = s$

Step 2

Subtract $676$ from both sides of the equation.

$- 16 t^{2} = s - 676$

Step 3

Divide each term in $- 16 t^{2} = s - 676$ by $- 16$ and simplify.

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Step 3.1

Divide each term in $- 16 t^{2} = s - 676$ by $- 16$ .

$\frac{- 16 t^{2}}{- 16} = \frac{s}{- 16} + \frac{- 676}{- 16}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 16$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 16 t^{2}}{- 16} = \frac{s}{- 16} + \frac{- 676}{- 16}$

Step 3.2.1.2

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

$t^{2} = \frac{s}{- 16} + \frac{- 676}{- 16}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$t^{2} = - \frac{s}{16} + \frac{- 676}{- 16}$

Step 3.3.1.2

Cancel the common factor of $- 676$ and $- 16$ .

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Step 3.3.1.2.1

Factor $- 4$ out of $- 676$ .

$t^{2} = - \frac{s}{16} + \frac{- 4 (169)}{- 16}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $- 4$ out of $- 16$ .

$t^{2} = - \frac{s}{16} + \frac{- 4 \cdot 169}{- 4 \cdot 4}$

Step 3.3.1.2.2.2

Cancel the common factor.

$t^{2} = - \frac{s}{16} + \frac{- 4 \cdot 169}{- 4 \cdot 4}$

Step 3.3.1.2.2.3

Rewrite the expression.

$t^{2} = - \frac{s}{16} + \frac{169}{4}$

Step 4

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

$t = \pm \sqrt{- \frac{s}{16} + \frac{169}{4}}$

Step 5

Simplify $\pm \sqrt{- \frac{s}{16} + \frac{169}{4}}$ .

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Step 5.1

To write $\frac{169}{4}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$t = \pm \sqrt{- \frac{s}{16} + \frac{169}{4} \cdot \frac{4}{4}}$

Step 5.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1

Multiply $\frac{169}{4}$ by $\frac{4}{4}$ .

$t = \pm \sqrt{- \frac{s}{16} + \frac{169 \cdot 4}{4 \cdot 4}}$

Step 5.2.2

Multiply $4$ by $4$ .

$t = \pm \sqrt{- \frac{s}{16} + \frac{169 \cdot 4}{16}}$

Step 5.3

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{- s + 169 \cdot 4}{16}}$

Step 5.4

Multiply $169$ by $4$ .

$t = \pm \sqrt{\frac{- s + 676}{16}}$

Step 5.5

Rewrite $\frac{- s + 676}{16}$ as ${(\frac{1}{4})}^{2} (- s + 676)$ .

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Step 5.5.1

Factor the perfect power $1^{2}$ out of $- s + 676$ .

$t = \pm \sqrt{\frac{1^{2} (- s + 676)}{16}}$

Step 5.5.2

Factor the perfect power $4^{2}$ out of $16$ .

$t = \pm \sqrt{\frac{1^{2} (- s + 676)}{4^{2} \cdot 1}}$

Step 5.5.3

Rearrange the fraction $\frac{1^{2} (- s + 676)}{4^{2} \cdot 1}$ .

$t = \pm \sqrt{{(\frac{1}{4})}^{2} (- s + 676)}$

Step 5.6

Pull terms out from under the radical.

$t = \pm \frac{1}{4} \sqrt{- s + 676}$

Step 5.7

Combine $\frac{1}{4}$ and $\sqrt{- s + 676}$ .

$t = \pm \frac{\sqrt{- s + 676}}{4}$

Step 6

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

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Step 6.1

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

$t = \frac{\sqrt{- s + 676}}{4}$

Step 6.2

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

$t = - \frac{\sqrt{- s + 676}}{4}$

Step 6.3

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

$t = \frac{\sqrt{- s + 676}}{4}$

$t = - \frac{\sqrt{- s + 676}}{4}$

$t = \frac{\sqrt{- s + 676}}{4}$

$t = - \frac{\sqrt{- s + 676}}{4}$