Pre-Algebra Examples

$4 x^{2} - 9 + 3 = 0$

Step 1

Add $- 9$ and $3$ .

$4 x^{2} - 6 = 0$

Step 2

Add $6$ to both sides of the equation.

$4 x^{2} = 6$

Step 3

Divide each term in $4 x^{2} = 6$ by $4$ and simplify.

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Step 3.1

Divide each term in $4 x^{2} = 6$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{6}{4}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $4$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{6}{4}$

Step 3.2.1.2

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

$x^{2} = \frac{6}{4}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $6$ and $4$ .

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Step 3.3.1.1

Factor $2$ out of $6$ .

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

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

Factor $2$ out of $4$ .

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

Step 3.3.1.2.2

Cancel the common factor.

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

Step 3.3.1.2.3

Rewrite the expression.

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

Step 4

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

$x = \pm \sqrt{\frac{3}{2}}$

Step 5

Simplify $\pm \sqrt{\frac{3}{2}}$ .

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Step 5.1

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

$x = \pm \frac{\sqrt{3}}{\sqrt{2}}$

Step 5.2

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 5.3

Combine and simplify the denominator.

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Step 5.3.1

Multiply $\frac{\sqrt{3}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.3.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.3.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 5.3.6

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

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Step 5.3.6.1

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

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

Step 5.3.6.2

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

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

Step 5.3.6.3

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

$x = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}}$

Step 5.3.6.4.2

Rewrite the expression.

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

Step 5.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3} \sqrt{2}}{2}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{3 \cdot 2}}{2}$

Step 5.4.2

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{6}}{2}$

Step 6

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

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Step 6.1

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

$x = \frac{\sqrt{6}}{2}$

Step 6.2

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

$x = - \frac{\sqrt{6}}{2}$

Step 6.3

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

$x = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{6}}{2}, - \frac{\sqrt{6}}{2}$

Decimal Form:

$x = 1.22474487 \dots, - 1.22474487 \dots$