Algebra Examples

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

Step 1

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

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

Step 2

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 3

Add the term to each side of the equation.

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

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

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

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

Step 4.1.1.2

One to any power is one.

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

Step 4.1.1.3

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

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

Step 4.2

Simplify the right side.

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Step 4.2.1

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

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

One to any power is one.

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

Step 4.2.1.1.3

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

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Combine the numerators over the common denominator.

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

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $4$ by $36$ .

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

Step 4.2.1.5.2

Add $144$ and $1$ .

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

Step 5

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

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

Step 6

Solve the equation for $x$ .

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Step 6.1

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

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the denominator.

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Step 6.2.2.1

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

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

Step 6.2.2.2

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

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

Step 6.3

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

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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