Pre-Algebra Examples

$\frac{3}{2} x^{2} - \frac{2}{3} x - \frac{4}{3} = 0$

Step 1

Simplify each term.

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Step 1.1

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

$\frac{3 x^{2}}{2} - \frac{2}{3} x - \frac{4}{3} = 0$

Step 1.2

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

$\frac{3 x^{2}}{2} - \frac{x \cdot 2}{3} - \frac{4}{3} = 0$

Step 1.3

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

$\frac{3 x^{2}}{2} - \frac{2 x}{3} - \frac{4}{3} = 0$

Step 2

Multiply through by the least common denominator $6$ , then simplify.

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Step 2.1

Apply the distributive property.

$6 (\frac{3 x^{2}}{2}) + 6 (- \frac{2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Factor $2$ out of $6$ .

$2 (3) (\frac{3 x^{2}}{2}) + 6 (- \frac{2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.1.2

Cancel the common factor.

$2 \cdot (3 (\frac{3 x^{2}}{2})) + 6 (- \frac{2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.1.3

Rewrite the expression.

$3 (3 x^{2}) + 6 (- \frac{2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.2

Multiply $3$ by $3$ .

$9 x^{2} + 6 (- \frac{2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.3

Cancel the common factor of $3$ .

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Step 2.2.3.1

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

$9 x^{2} + 6 (\frac{- 2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.3.2

Factor $3$ out of $6$ .

$9 x^{2} + 3 (2) (\frac{- 2 x}{3}) + 6 (- \frac{4}{3}) = 0$

Step 2.2.3.3

Cancel the common factor.

$9 x^{2} + 3 \cdot (2 (\frac{- 2 x}{3})) + 6 (- \frac{4}{3}) = 0$

Step 2.2.3.4

Rewrite the expression.

$9 x^{2} + 2 (- 2 x) + 6 (- \frac{4}{3}) = 0$

Step 2.2.4

Multiply $- 2$ by $2$ .

$9 x^{2} - 4 x + 6 (- \frac{4}{3}) = 0$

Step 2.2.5

Cancel the common factor of $3$ .

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Step 2.2.5.1

Move the leading negative in $- \frac{4}{3}$ into the numerator.

$9 x^{2} - 4 x + 6 (\frac{- 4}{3}) = 0$

Step 2.2.5.2

Factor $3$ out of $6$ .

$9 x^{2} - 4 x + 3 (2) (\frac{- 4}{3}) = 0$

Step 2.2.5.3

Cancel the common factor.

$9 x^{2} - 4 x + 3 \cdot (2 (\frac{- 4}{3})) = 0$

Step 2.2.5.4

Rewrite the expression.

$9 x^{2} - 4 x + 2 \cdot - 4 = 0$

Step 2.2.6

Multiply $2$ by $- 4$ .

$9 x^{2} - 4 x - 8 = 0$

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 $a = 9$ , $b = - 4$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

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

Step 5.1.2

Multiply $- 4 \cdot 9 \cdot - 8$ .

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Step 5.1.2.1

Multiply $- 4$ by $9$ .

$x = \frac{4 \pm \sqrt{16 - 36 \cdot - 8}}{2 \cdot 9}$

Step 5.1.2.2

Multiply $- 36$ by $- 8$ .

$x = \frac{4 \pm \sqrt{16 + 288}}{2 \cdot 9}$

Step 5.1.3

Add $16$ and $288$ .

$x = \frac{4 \pm \sqrt{304}}{2 \cdot 9}$

Step 5.1.4

Rewrite $304$ as $4^{2} \cdot 19$ .

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Step 5.1.4.1

Factor $16$ out of $304$ .

$x = \frac{4 \pm \sqrt{16 (19)}}{2 \cdot 9}$

Step 5.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 19}}{2 \cdot 9}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{19}}{2 \cdot 9}$

Step 5.2

Multiply $2$ by $9$ .

$x = \frac{4 \pm 4 \sqrt{19}}{18}$

Step 5.3

Simplify $\frac{4 \pm 4 \sqrt{19}}{18}$ .

$x = \frac{2 \pm 2 \sqrt{19}}{9}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{2 + 2 \sqrt{19}}{9}, \frac{2 - 2 \sqrt{19}}{9}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 + 2 \sqrt{19}}{9}, \frac{2 - 2 \sqrt{19}}{9}$

Decimal Form:

$x = 1.19086643 \dots, - 0.74642198 \dots$