Pre-Algebra Examples

$3 x^{2} + 28 = 0$

Step 1

Subtract $28$ from both sides of the equation.

$3 x^{2} = - 28$

Step 2

Divide each term in $3 x^{2} = - 28$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 x^{2} = - 28$ by $3$ .

$\frac{3 x^{2}}{3} = \frac{- 28}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{- 28}{3}$

Step 2.2.1.2

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

$x^{2} = \frac{- 28}{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{28}{3}$

Step 3

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

$x = \pm \sqrt{- \frac{28}{3}}$

Step 4

Simplify $\pm \sqrt{- \frac{28}{3}}$ .

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Step 4.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{28}{3}}$

Step 4.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{28}{3}}$

Step 4.3

Rewrite $\sqrt{\frac{28}{3}}$ as $\frac{\sqrt{28}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{28}}{\sqrt{3}}$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Rewrite $28$ as $2^{2} \cdot 7$ .

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Step 4.4.1.1

Factor $4$ out of $28$ .

$x = \pm i \frac{\sqrt{4 (7)}}{\sqrt{3}}$

Step 4.4.1.2

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

$x = \pm i \frac{\sqrt{2^{2} \cdot 7}}{\sqrt{3}}$

Step 4.4.2

Pull terms out from under the radical.

$x = \pm i \frac{2 \sqrt{7}}{\sqrt{3}}$

Step 4.5

Multiply $\frac{2 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{2 \sqrt{7}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.6

Combine and simplify the denominator.

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Step 4.6.1

Multiply $\frac{2 \sqrt{7}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.6.4

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

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.6.5

Add $1$ and $1$ .

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.6.6

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

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Step 4.6.6.1

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

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.6.6.2

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

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.6.6.3

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

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.6.4

Cancel the common factor of $2$ .

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Step 4.6.6.4.1

Cancel the common factor.

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.6.6.4.2

Rewrite the expression.

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{3^{1}}$

Step 4.6.6.5

Evaluate the exponent.

$x = \pm i \frac{2 \sqrt{7} \sqrt{3}}{3}$

Step 4.7

Simplify the numerator.

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Step 4.7.1

Combine using the product rule for radicals.

$x = \pm i \frac{2 \sqrt{3 \cdot 7}}{3}$

Step 4.7.2

Multiply $3$ by $7$ .

$x = \pm i \frac{2 \sqrt{21}}{3}$

Step 4.8

Combine fractions.

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Step 4.8.1

Combine $i$ and $\frac{2 \sqrt{21}}{3}$ .

$x = \pm \frac{i (2 \sqrt{21})}{3}$

Step 4.8.2

Move $2$ to the left of $i$ .

$x = \pm \frac{2 i \sqrt{21}}{3}$

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.

$x = \frac{2 i \sqrt{21}}{3}$

Step 5.2

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

$x = - \frac{2 i \sqrt{21}}{3}$

Step 5.3

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

$x = \frac{2 i \sqrt{21}}{3}, - \frac{2 i \sqrt{21}}{3}$