Pre-Algebra Examples

$- 3 \cdot - 3 = 3^{3} - 3 x^{2} + x$

Step 1

Rewrite the equation as $3^{3} - 3 x^{2} + x = - 3 \cdot - 3$ .

$3^{3} - 3 x^{2} + x = - 3 \cdot - 3$

Step 2

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

$27 - 3 x^{2} + x = - 3 \cdot - 3$

Step 3

Multiply $- 3$ by $- 3$ .

$27 - 3 x^{2} + x = 9$

Step 4

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

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Step 4.1

Subtract $9$ from both sides of the equation.

$27 - 3 x^{2} + x - 9 = 0$

Step 4.2

Subtract $9$ from $27$ .

$- 3 x^{2} + x + 18 = 0$

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 $a = - 3$ , $b = 1$ , and $c = 18$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 3 \cdot 18)}}{2 \cdot - 3}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 3 \cdot 18}}{2 \cdot - 3}$

Step 7.1.2

Multiply $- 4 \cdot - 3 \cdot 18$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12 \cdot 18}}{2 \cdot - 3}$

Step 7.1.2.2

Multiply $12$ by $18$ .

$x = \frac{- 1 \pm \sqrt{1 + 216}}{2 \cdot - 3}$

Step 7.1.3

Add $1$ and $216$ .

$x = \frac{- 1 \pm \sqrt{217}}{2 \cdot - 3}$

Step 7.2

Multiply $2$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{217}}{- 6}$

Step 7.3

Simplify $\frac{- 1 \pm \sqrt{217}}{- 6}$ .

$x = \frac{1 \pm \sqrt{217}}{6}$

Step 8

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 8.1

Simplify the numerator.

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Step 8.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 3 \cdot 18}}{2 \cdot - 3}$

Step 8.1.2

Multiply $- 4 \cdot - 3 \cdot 18$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12 \cdot 18}}{2 \cdot - 3}$

Step 8.1.2.2

Multiply $12$ by $18$ .

$x = \frac{- 1 \pm \sqrt{1 + 216}}{2 \cdot - 3}$

Step 8.1.3

Add $1$ and $216$ .

$x = \frac{- 1 \pm \sqrt{217}}{2 \cdot - 3}$

Step 8.2

Multiply $2$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{217}}{- 6}$

Step 8.3

Simplify $\frac{- 1 \pm \sqrt{217}}{- 6}$ .

$x = \frac{1 \pm \sqrt{217}}{6}$

Step 8.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{217}}{6}$

Step 9

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 9.1

Simplify the numerator.

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Step 9.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 3 \cdot 18}}{2 \cdot - 3}$

Step 9.1.2

Multiply $- 4 \cdot - 3 \cdot 18$ .

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Step 9.1.2.1

Multiply $- 4$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{1 + 12 \cdot 18}}{2 \cdot - 3}$

Step 9.1.2.2

Multiply $12$ by $18$ .

$x = \frac{- 1 \pm \sqrt{1 + 216}}{2 \cdot - 3}$

Step 9.1.3

Add $1$ and $216$ .

$x = \frac{- 1 \pm \sqrt{217}}{2 \cdot - 3}$

Step 9.2

Multiply $2$ by $- 3$ .

$x = \frac{- 1 \pm \sqrt{217}}{- 6}$

Step 9.3

Simplify $\frac{- 1 \pm \sqrt{217}}{- 6}$ .

$x = \frac{1 \pm \sqrt{217}}{6}$

Step 9.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{217}}{6}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{217}}{6}, \frac{1 - \sqrt{217}}{6}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{217}}{6}, \frac{1 - \sqrt{217}}{6}$

Decimal Form:

$x = 2.62181997 \dots, - 2.28848664 \dots$