Precalculus Examples

$x^{2} - \frac{2}{3} x = - \frac{2}{9}$

Step 1

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

Tap for more steps...

Step 1.1

Simplify the left side.

Tap for more steps...

Step 1.1.1

Simplify each term.

Tap for more steps...

Step 1.1.1.1

Combine $x$ and $\frac{2}{3}$ .

$x^{2} - \frac{x \cdot 2}{3} = - \frac{2}{9}$

Step 1.1.1.2

Move $2$ to the left of $x$ .

$x^{2} - \frac{2 x}{3} = - \frac{2}{9}$

Step 1.2

Add $\frac{2}{9}$ to both sides of the equation.

$x^{2} - \frac{2 x}{3} + \frac{2}{9} = 0$

Step 2

Multiply through by the least common denominator $9$ , then simplify.

Tap for more steps...

Step 2.1

Apply the distributive property.

$9 x^{2} + 9 (- \frac{2 x}{3}) + 9 (\frac{2}{9}) = 0$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1

Move the leading negative in $- \frac{2 x}{3}$ into the numerator.

$9 x^{2} + 9 (\frac{- 2 x}{3}) + 9 (\frac{2}{9}) = 0$

Step 2.2.1.2

Factor $3$ out of $9$ .

$9 x^{2} + 3 (3) (\frac{- 2 x}{3}) + 9 (\frac{2}{9}) = 0$

Step 2.2.1.3

Cancel the common factor.

$9 x^{2} + 3 \cdot (3 (\frac{- 2 x}{3})) + 9 (\frac{2}{9}) = 0$

Step 2.2.1.4

Rewrite the expression.

$9 x^{2} + 3 (- 2 x) + 9 (\frac{2}{9}) = 0$

Step 2.2.2

Multiply $- 2$ by $3$ .

$9 x^{2} - 6 x + 9 (\frac{2}{9}) = 0$

Step 2.2.3

Cancel the common factor of $9$ .

Tap for more steps...

Step 2.2.3.1

Cancel the common factor.

$9 x^{2} - 6 x + 9 (\frac{2}{9}) = 0$

Step 2.2.3.2

Rewrite the expression.

$9 x^{2} - 6 x + 2 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 9$ , $b = - 6$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (9 \cdot 2)}}{2 \cdot 9}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 9 \cdot 2}}{2 \cdot 9}$

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot 2$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{6 \pm \sqrt{36 - 36 \cdot 2}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $2$ .

$x = \frac{6 \pm \sqrt{36 - 72}}{2 \cdot 9}$

Step 5.1.3

Subtract $72$ from $36$ .

$x = \frac{6 \pm \sqrt{- 36}}{2 \cdot 9}$

Step 5.1.4

Rewrite $- 36$ as $- 1 (36)$ .

$x = \frac{6 \pm \sqrt{- 1 \cdot 36}}{2 \cdot 9}$

Step 5.1.5

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \frac{6 \pm \sqrt{- 1} \cdot \sqrt{36}}{2 \cdot 9}$

Step 5.1.6

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

$x = \frac{6 \pm i \cdot \sqrt{36}}{2 \cdot 9}$

Step 5.1.7

Rewrite $36$ as $6^{2}$ .

$x = \frac{6 \pm i \cdot \sqrt{6^{2}}}{2 \cdot 9}$

Step 5.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{6 \pm i \cdot 6}{2 \cdot 9}$

Step 5.1.9

Move $6$ to the left of $i$ .

$x = \frac{6 \pm 6 i}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{6 \pm 6 i}{18}$

Step 5.3

Simplify $\frac{6 \pm 6 i}{18}$ .

$x = \frac{1 \pm i}{3}$