Algebra Examples

$r^{2} - \frac{3 r}{5} + \frac{2}{5} = 0$

Step 1

Multiply through by the least common denominator $5$ , then simplify.

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Step 1.1

Apply the distributive property.

$5 r^{2} + 5 (- \frac{3 r}{5}) + 5 (\frac{2}{5}) = 0$

Step 1.2

Simplify.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Move the leading negative in $- \frac{3 r}{5}$ into the numerator.

$5 r^{2} + 5 (\frac{- 3 r}{5}) + 5 (\frac{2}{5}) = 0$

Step 1.2.1.2

Cancel the common factor.

$5 r^{2} + 5 (\frac{- 3 r}{5}) + 5 (\frac{2}{5}) = 0$

Step 1.2.1.3

Rewrite the expression.

$5 r^{2} - 3 r + 5 (\frac{2}{5}) = 0$

Step 1.2.2

Cancel the common factor of $5$ .

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Step 1.2.2.1

Cancel the common factor.

$5 r^{2} - 3 r + 5 (\frac{2}{5}) = 0$

Step 1.2.2.2

Rewrite the expression.

$5 r^{2} - 3 r + 2 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 5$ , $b = - 3$ , and $c = 2$ into the quadratic formula and solve for $r$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (5 \cdot 2)}}{2 \cdot 5}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 3$ to the power of $2$ .

$r = \frac{3 \pm \sqrt{9 - 4 \cdot 5 \cdot 2}}{2 \cdot 5}$

Step 4.1.2

Multiply $- 4 \cdot 5 \cdot 2$ .

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Step 4.1.2.1

Multiply $- 4$ by $5$ .

$r = \frac{3 \pm \sqrt{9 - 20 \cdot 2}}{2 \cdot 5}$

Step 4.1.2.2

Multiply $- 20$ by $2$ .

$r = \frac{3 \pm \sqrt{9 - 40}}{2 \cdot 5}$

Step 4.1.3

Subtract $40$ from $9$ .

$r = \frac{3 \pm \sqrt{- 31}}{2 \cdot 5}$

Step 4.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$r = \frac{3 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 5}$

Step 4.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$r = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 5}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$r = \frac{3 \pm i \sqrt{31}}{2 \cdot 5}$

Step 4.2

Multiply $2$ by $5$ .

$r = \frac{3 \pm i \sqrt{31}}{10}$