Algebra Examples

$r = r_{0} + \frac{1}{5} a t^{2}$

Step 1

Rewrite the equation as $r_{0} + \frac{1}{5} \cdot (a t^{2}) = r$ .

$r_{0} + \frac{1}{5} \cdot (a t^{2}) = r$

Step 2

Multiply $\frac{1}{5} (a t^{2})$ .

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Step 2.1

Combine $a$ and $\frac{1}{5}$ .

$r_{0} + \frac{a}{5} t^{2} = r$

Step 2.2

Combine $\frac{a}{5}$ and $t^{2}$ .

$r_{0} + \frac{a t^{2}}{5} = r$

Step 3

Subtract $r_{0}$ from both sides of the equation.

$\frac{a t^{2}}{5} = r - r_{0}$

Step 4

Multiply both sides of the equation by $5$ .

$5 \frac{a t^{2}}{5} = 5 (r - r_{0})$

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $5$ .

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Step 5.1.1.1

Cancel the common factor.

$5 \frac{a t^{2}}{5} = 5 (r - r_{0})$

Step 5.1.1.2

Rewrite the expression.

$a t^{2} = 5 (r - r_{0})$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $5 (r - r_{0})$ .

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Step 5.2.1.1

Apply the distributive property.

$a t^{2} = 5 r + 5 (- r_{0})$

Step 5.2.1.2

Multiply $- 1$ by $5$ .

$a t^{2} = 5 r - 5 r_{0}$

Step 6

Divide each term in $a t^{2} = 5 r - 5 r_{0}$ by $a$ and simplify.

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Step 6.1

Divide each term in $a t^{2} = 5 r - 5 r_{0}$ by $a$ .

$\frac{a t^{2}}{a} = \frac{5 r}{a} + \frac{- 5 r_{0}}{a}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $a$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{a t^{2}}{a} = \frac{5 r}{a} + \frac{- 5 r_{0}}{a}$

Step 6.2.1.2

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

$t^{2} = \frac{5 r}{a} + \frac{- 5 r_{0}}{a}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

$t^{2} = \frac{5 r}{a} - \frac{5 r_{0}}{a}$

Step 7

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

$t = \pm \sqrt{\frac{5 r}{a} - \frac{5 r_{0}}{a}}$

Step 8

Simplify $\pm \sqrt{\frac{5 r}{a} - \frac{5 r_{0}}{a}}$ .

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Step 8.1

Combine the numerators over the common denominator.

$t = \pm \sqrt{\frac{5 r - 5 r_{0}}{a}}$

Step 8.2

Factor $5$ out of $5 r - 5 r_{0}$ .

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Step 8.2.1

Factor $5$ out of $5 r$ .

$t = \pm \sqrt{\frac{5 (r) - 5 r_{0}}{a}}$

Step 8.2.2

Factor $5$ out of $- 5 r_{0}$ .

$t = \pm \sqrt{\frac{5 (r) + 5 (- r_{0})}{a}}$

Step 8.2.3

Factor $5$ out of $5 (r) + 5 (- r_{0})$ .

$t = \pm \sqrt{\frac{5 (r - r_{0})}{a}}$

Step 8.3

Rewrite $\sqrt{\frac{5 (r - r_{0})}{a}}$ as $\frac{\sqrt{5 (r - r_{0})}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})}}{\sqrt{a}}$

Step 8.4

Multiply $\frac{\sqrt{5 (r - r_{0})}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})}}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}}$

Step 8.5

Combine and simplify the denominator.

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Step 8.5.1

Multiply $\frac{\sqrt{5 (r - r_{0})}}{\sqrt{a}}$ by $\frac{\sqrt{a}}{\sqrt{a}}$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{\sqrt{a} \sqrt{a}}$

Step 8.5.2

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{{\sqrt{a}}^{1} \sqrt{a}}$

Step 8.5.3

Raise $\sqrt{a}$ to the power of $1$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{{\sqrt{a}}^{1} {\sqrt{a}}^{1}}$

Step 8.5.4

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

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{{\sqrt{a}}^{1 + 1}}$

Step 8.5.5

Add $1$ and $1$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{{\sqrt{a}}^{2}}$

Step 8.5.6

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

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Step 8.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a}$ as $a^{\frac{1}{2}}$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{{(a^{\frac{1}{2}})}^{2}}$

Step 8.5.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{a^{\frac{1}{2} \cdot 2}}$

Step 8.5.6.3

Combine $\frac{1}{2}$ and $2$ .

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{a^{\frac{2}{2}}}$

Step 8.5.6.4

Cancel the common factor of $2$ .

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Step 8.5.6.4.1

Cancel the common factor.

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{a^{\frac{2}{2}}}$

Step 8.5.6.4.2

Rewrite the expression.

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{a^{1}}$

Step 8.5.6.5

Simplify.

$t = \pm \frac{\sqrt{5 (r - r_{0})} \sqrt{a}}{a}$

Step 8.6

Combine using the product rule for radicals.

$t = \pm \frac{\sqrt{5 (r - r_{0}) a}}{a}$

Step 8.7

Reorder factors in $\pm \frac{\sqrt{5 (r - r_{0}) a}}{a}$ .

$t = \pm \frac{\sqrt{5 a (r - r_{0})}}{a}$

Step 9

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

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Step 9.1

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

$t = \frac{\sqrt{5 a (r - r_{0})}}{a}$

Step 9.2

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

$t = - \frac{\sqrt{5 a (r - r_{0})}}{a}$

Step 9.3

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

$t = \frac{\sqrt{5 a (r - r_{0})}}{a}$

$t = - \frac{\sqrt{5 a (r - r_{0})}}{a}$

$t = \frac{\sqrt{5 a (r - r_{0})}}{a}$

$t = - \frac{\sqrt{5 a (r - r_{0})}}{a}$