Algebra Examples

$\frac{1}{4} x^{2} + 11 = 38$

Step 1

Subtract $38$ from both sides of the equation.

$\frac{1}{4} x^{2} + 11 - 38 = 0$

Step 2

Simplify $\frac{1}{4} x^{2} + 11 - 38$ .

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Step 2.1

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + 11 - 38 = 0$

Step 2.2

Subtract $38$ from $11$ .

$\frac{x^{2}}{4} - 27 = 0$

Step 3

To write $- 27$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{x^{2}}{4} - 27 \cdot \frac{4}{4} = 0$

Step 4

Combine $- 27$ and $\frac{4}{4}$ .

$\frac{x^{2}}{4} + \frac{- 27 \cdot 4}{4} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{x^{2} - 27 \cdot 4}{4} = 0$

Step 6

Multiply $- 27$ by $4$ .

$\frac{x^{2} - 108}{4} = 0$

Step 7

Set the numerator equal to zero.

$x^{2} - 108 = 0$

Step 8

Solve the equation for $x$ .

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Step 8.1

Add $108$ to both sides of the equation.

$x^{2} = 108$

Step 8.2

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

$x = \pm \sqrt{108}$

Step 8.3

Simplify $\pm \sqrt{108}$ .

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Step 8.3.1

Rewrite $108$ as $6^{2} \cdot 3$ .

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Step 8.3.1.1

Factor $36$ out of $108$ .

$x = \pm \sqrt{36 (3)}$

Step 8.3.1.2

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

$x = \pm \sqrt{6^{2} \cdot 3}$

Step 8.3.2

Pull terms out from under the radical.

$x = \pm 6 \sqrt{3}$

Step 8.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 8.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 6 \sqrt{3}$

Step 8.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 6 \sqrt{3}$

Step 8.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 6 \sqrt{3}, - 6 \sqrt{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 6 \sqrt{3}, - 6 \sqrt{3}$

Decimal Form:

$x = 10.39230484 \dots, - 10.39230484 \dots$