Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

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

Step 1.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 1.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 1.2.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 1.2.4

Cancel the common factor of $2$ .

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Step 1.2.4.1

Cancel the common factor.

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

Step 1.2.4.2

Rewrite the expression.

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

Step 1.2.5

Evaluate the exponent.

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

Step 1.3

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

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

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Subtract $\frac{3}{4}$ from both sides of the equation.

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

Step 2.2

Write $1$ as a fraction with a common denominator.

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

Step 2.3

Combine the numerators over the common denominator.

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

Step 2.4

Subtract $3$ from $4$ .

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

Step 3

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

$x = \pm \sqrt{\frac{1}{4}}$

Step 4

Simplify $\pm \sqrt{\frac{1}{4}}$ .

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Step 4.1

Rewrite $\sqrt{\frac{1}{4}}$ as $\frac{\sqrt{1}}{\sqrt{4}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{4}}$

Step 4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{4}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 4.3.2

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

$x = \pm \frac{1}{2}$

Step 5

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

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Step 5.1

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

$x = \frac{1}{2}$

Step 5.2

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

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

Step 5.3

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

$x = \frac{1}{2}, - \frac{1}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{2}, - \frac{1}{2}$

Decimal Form:

$x = 0.5, - 0.5$