Solve by Completing the Square 3x^2+x-1/3=0

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

Step 1

Add $\frac{1}{3}$ to both sides of the equation.

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

Step 2

Divide each term in $3 x^{2} + x = \frac{1}{3}$ by $3$ and simplify.

Divide each term in $3 x^{2} + x = \frac{1}{3}$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x^{2} + \frac{x}{3} = \frac{1}{3} \cdot \frac{1}{3}$

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$x^{2} + \frac{x}{3} = \frac{1}{3 \cdot 3}$

Multiply $3$ by $3$ .

$x^{2} + \frac{x}{3} = \frac{1}{9}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{6})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{x}{3} + {(\frac{1}{6})}^{2} = \frac{1}{9} + {(\frac{1}{6})}^{2}$

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

Apply the product rule to $\frac{1}{6}$ .

$x^{2} + \frac{x}{3} + \frac{1^{2}}{6^{2}} = \frac{1}{9} + {(\frac{1}{6})}^{2}$

One to any power is one.

$x^{2} + \frac{x}{3} + \frac{1}{6^{2}} = \frac{1}{9} + {(\frac{1}{6})}^{2}$

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

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{1}{9} + {(\frac{1}{6})}^{2}$

Simplify the right side.

Simplify $\frac{1}{9} + {(\frac{1}{6})}^{2}$ .

Simplify each term.

Apply the product rule to $\frac{1}{6}$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{1}{9} + \frac{1^{2}}{6^{2}}$

One to any power is one.

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{1}{9} + \frac{1}{6^{2}}$

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

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{1}{9} + \frac{1}{36}$

To write $\frac{1}{9}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{1}{9} \cdot \frac{4}{4} + \frac{1}{36}$

Write each expression with a common denominator of $36$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{9}$ by $\frac{4}{4}$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{4}{9 \cdot 4} + \frac{1}{36}$

Multiply $9$ by $4$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{4}{36} + \frac{1}{36}$

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{4 + 1}{36}$

Add $4$ and $1$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{5}{36}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{1}{6})}^{2}$ .

${(x + \frac{1}{6})}^{2} = \frac{5}{36}$

Step 7

Solve the equation for $x$ .

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + \frac{1}{6} = \pm \sqrt{\frac{5}{36}}$

Simplify $\pm \sqrt{\frac{5}{36}}$ .

Rewrite $\sqrt{\frac{5}{36}}$ as $\frac{\sqrt{5}}{\sqrt{36}}$ .

$x + \frac{1}{6} = \pm \frac{\sqrt{5}}{\sqrt{36}}$

Simplify the denominator.

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

$x + \frac{1}{6} = \pm \frac{\sqrt{5}}{\sqrt{6^{2}}}$

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

$x + \frac{1}{6} = \pm \frac{\sqrt{5}}{6}$

Subtract $\frac{1}{6}$ from both sides of the equation.

$x = \pm \frac{\sqrt{5}}{6} - \frac{1}{6}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{5}}{6} - \frac{1}{6}$

Decimal Form:

$x = 0.20601132 \dots, - 0.53934466 \dots$

Solve by Completing the Square 3x^2+x-1/3=0

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