Solve by Completing the Square x^2+1/3x=4

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

Combine $\frac{1}{3}$ and $x$ .

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

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

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

Add the term to each side of the equation.

$x^{2} + \frac{x}{3} + {(\frac{1}{6})}^{2} = 4 + {(\frac{1}{6})}^{2}$

Simplify the equation.

Simplify the left side.

Simplify each term.

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

$x^{2} + \frac{x}{3} + \frac{1^{2}}{6^{2}} = 4 + {(\frac{1}{6})}^{2}$

One to any power is one.

$x^{2} + \frac{x}{3} + \frac{1}{6^{2}} = 4 + {(\frac{1}{6})}^{2}$

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

$x^{2} + \frac{x}{3} + \frac{1}{36} = 4 + {(\frac{1}{6})}^{2}$

Simplify the right side.

Simplify $4 + {(\frac{1}{6})}^{2}$ .

Simplify each term.

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

$x^{2} + \frac{x}{3} + \frac{1}{36} = 4 + \frac{1^{2}}{6^{2}}$

One to any power is one.

$x^{2} + \frac{x}{3} + \frac{1}{36} = 4 + \frac{1}{6^{2}}$

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

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

To write $4$ as a fraction with a common denominator, multiply by $\frac{36}{36}$ .

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

Combine $4$ and $\frac{36}{36}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $4$ by $36$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{144 + 1}{36}$

Add $144$ and $1$ .

$x^{2} + \frac{x}{3} + \frac{1}{36} = \frac{145}{36}$

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

${(x + \frac{1}{6})}^{2} = \frac{145}{36}$

Solve the equation for $x$ .

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

$x + \frac{1}{6} = \pm \sqrt{\frac{145}{36}}$

Simplify $\pm \sqrt{\frac{145}{36}}$ .

Rewrite $\sqrt{\frac{145}{36}}$ as $\frac{\sqrt{145}}{\sqrt{36}}$ .

$x + \frac{1}{6} = \pm \frac{\sqrt{145}}{\sqrt{36}}$

Simplify the denominator.

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

$x + \frac{1}{6} = \pm \frac{\sqrt{145}}{\sqrt{6^{2}}}$

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

$x + \frac{1}{6} = \pm \frac{\sqrt{145}}{6}$

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

$x = \pm \frac{\sqrt{145}}{6} - \frac{1}{6}$

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{145}}{6} - \frac{1}{6}$

Decimal Form:

$x = 1.84026576 \dots, - 2.17359909 \dots$

Solve by Completing the Square x^2+1/3x=4

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