Calculus Examples

$\frac{2}{3} x^{2} - x = \frac{1}{3}$

Step 1

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

$\frac{2 x^{2}}{3} - x = \frac{1}{3}$

Step 2

Subtract $\frac{1}{3}$ from both sides of the equation.

$\frac{2 x^{2}}{3} - x - \frac{1}{3} = 0$

Step 3

Multiply through by the least common denominator $3$ , then simplify.

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Step 3.1

Apply the distributive property.

$3 (\frac{2 x^{2}}{3}) + 3 (- x) + 3 (- \frac{1}{3}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $3$ .

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Step 3.2.1.1

Cancel the common factor.

$3 (\frac{2 x^{2}}{3}) + 3 (- x) + 3 (- \frac{1}{3}) = 0$

Step 3.2.1.2

Rewrite the expression.

$2 x^{2} + 3 (- x) + 3 (- \frac{1}{3}) = 0$

Step 3.2.2

Multiply $- 1$ by $3$ .

$2 x^{2} - 3 x + 3 (- \frac{1}{3}) = 0$

Step 3.2.3

Cancel the common factor of $3$ .

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Step 3.2.3.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

$2 x^{2} - 3 x + 3 (\frac{- 1}{3}) = 0$

Step 3.2.3.2

Cancel the common factor.

$2 x^{2} - 3 x + 3 (\frac{- 1}{3}) = 0$

Step 3.2.3.3

Rewrite the expression.

$2 x^{2} - 3 x - 1 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 2$ , $b = - 3$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot - 1$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{3 \pm \sqrt{9 - 8 \cdot - 1}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $- 1$ .

$x = \frac{3 \pm \sqrt{9 + 8}}{2 \cdot 2}$

Step 6.1.3

Add $9$ and $8$ .

$x = \frac{3 \pm \sqrt{17}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{3 \pm \sqrt{17}}{4}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{17}}{4}, \frac{3 - \sqrt{17}}{4}$

Decimal Form:

$x = 1.78077640 \dots, - 0.28077640 \dots$