Basic Math Examples

$r^{2} + \frac{1}{2} r + \frac{1}{16} = \frac{10}{16}$

Step 1

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

$r^{2} + \frac{r}{2} + \frac{1}{16} = \frac{10}{16}$

Step 2

Cancel the common factor of $10$ and $16$ .

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Step 2.1

Factor $2$ out of $10$ .

$r^{2} + \frac{r}{2} + \frac{1}{16} = \frac{2 (5)}{16}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $2$ out of $16$ .

$r^{2} + \frac{r}{2} + \frac{1}{16} = \frac{2 \cdot 5}{2 \cdot 8}$

Step 2.2.2

Cancel the common factor.

$r^{2} + \frac{r}{2} + \frac{1}{16} = \frac{2 \cdot 5}{2 \cdot 8}$

Step 2.2.3

Rewrite the expression.

$r^{2} + \frac{r}{2} + \frac{1}{16} = \frac{5}{8}$

Step 3

Move all terms to the left side of the equation and simplify.

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Step 3.1

Subtract $\frac{5}{8}$ from both sides of the equation.

$r^{2} + \frac{r}{2} + \frac{1}{16} - \frac{5}{8} = 0$

Step 3.2

Simplify $r^{2} + \frac{r}{2} + \frac{1}{16} - \frac{5}{8}$ .

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Step 3.2.1

To write $- \frac{5}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$r^{2} + \frac{r}{2} + \frac{1}{16} - \frac{5}{8} \cdot \frac{2}{2} = 0$

Step 3.2.2

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

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Step 3.2.2.1

Multiply $\frac{5}{8}$ by $\frac{2}{2}$ .

$r^{2} + \frac{r}{2} + \frac{1}{16} - \frac{5 \cdot 2}{8 \cdot 2} = 0$

Step 3.2.2.2

Multiply $8$ by $2$ .

$r^{2} + \frac{r}{2} + \frac{1}{16} - \frac{5 \cdot 2}{16} = 0$

Step 3.2.3

Combine the numerators over the common denominator.

$r^{2} + \frac{r}{2} + \frac{1 - 5 \cdot 2}{16} = 0$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $- 5$ by $2$ .

$r^{2} + \frac{r}{2} + \frac{1 - 10}{16} = 0$

Step 3.2.4.2

Subtract $10$ from $1$ .

$r^{2} + \frac{r}{2} + \frac{- 9}{16} = 0$

Step 3.2.5

Move the negative in front of the fraction.

$r^{2} + \frac{r}{2} - \frac{9}{16} = 0$

Step 4

Multiply through by the least common denominator $16$ , then simplify.

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Step 4.1

Apply the distributive property.

$16 r^{2} + 16 (\frac{r}{2}) + 16 (- \frac{9}{16}) = 0$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Factor $2$ out of $16$ .

$16 r^{2} + 2 (8) (\frac{r}{2}) + 16 (- \frac{9}{16}) = 0$

Step 4.2.1.2

Cancel the common factor.

$16 r^{2} + 2 \cdot (8 (\frac{r}{2})) + 16 (- \frac{9}{16}) = 0$

Step 4.2.1.3

Rewrite the expression.

$16 r^{2} + 8 r + 16 (- \frac{9}{16}) = 0$

Step 4.2.2

Cancel the common factor of $16$ .

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Step 4.2.2.1

Move the leading negative in $- \frac{9}{16}$ into the numerator.

$16 r^{2} + 8 r + 16 (\frac{- 9}{16}) = 0$

Step 4.2.2.2

Cancel the common factor.

$16 r^{2} + 8 r + 16 (\frac{- 9}{16}) = 0$

Step 4.2.2.3

Rewrite the expression.

$16 r^{2} + 8 r - 9 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 16$ , $b = 8$ , and $c = - 9$ into the quadratic formula and solve for $r$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (16 \cdot - 9)}}{2 \cdot 16}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $8$ to the power of $2$ .

$r = \frac{- 8 \pm \sqrt{64 - 4 \cdot 16 \cdot - 9}}{2 \cdot 16}$

Step 7.1.2

Multiply $- 4 \cdot 16 \cdot - 9$ .

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Step 7.1.2.1

Multiply $- 4$ by $16$ .

$r = \frac{- 8 \pm \sqrt{64 - 64 \cdot - 9}}{2 \cdot 16}$

Step 7.1.2.2

Multiply $- 64$ by $- 9$ .

$r = \frac{- 8 \pm \sqrt{64 + 576}}{2 \cdot 16}$

Step 7.1.3

Add $64$ and $576$ .

$r = \frac{- 8 \pm \sqrt{640}}{2 \cdot 16}$

Step 7.1.4

Rewrite $640$ as $8^{2} \cdot 10$ .

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Step 7.1.4.1

Factor $64$ out of $640$ .

$r = \frac{- 8 \pm \sqrt{64 (10)}}{2 \cdot 16}$

Step 7.1.4.2

Rewrite $64$ as $8^{2}$ .

$r = \frac{- 8 \pm \sqrt{8^{2} \cdot 10}}{2 \cdot 16}$

Step 7.1.5

Pull terms out from under the radical.

$r = \frac{- 8 \pm 8 \sqrt{10}}{2 \cdot 16}$

Step 7.2

Multiply $2$ by $16$ .

$r = \frac{- 8 \pm 8 \sqrt{10}}{32}$

Step 7.3

Simplify $\frac{- 8 \pm 8 \sqrt{10}}{32}$ .

$r = \frac{- 1 \pm \sqrt{10}}{4}$

Step 8

The final answer is the combination of both solutions.

$r = - \frac{1 - \sqrt{10}}{4}, - \frac{1 + \sqrt{10}}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$r = - \frac{1 - \sqrt{10}}{4}, - \frac{1 + \sqrt{10}}{4}$

Decimal Form:

$r = 0.54056941 \dots, - 1.04056941 \dots$