Pre-Algebra Examples

$38 x^{2} + 4 x + 13 = 27 x^{2} + 4 x + 4$

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $27 x^{2}$ from both sides of the equation.

$38 x^{2} + 4 x + 13 - 27 x^{2} = 4 x + 4$

Step 1.2

Subtract $4 x$ from both sides of the equation.

$38 x^{2} + 4 x + 13 - 27 x^{2} - 4 x = 4$

Step 1.3

Combine the opposite terms in $38 x^{2} + 4 x + 13 - 27 x^{2} - 4 x$ .

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Step 1.3.1

Subtract $4 x$ from $4 x$ .

$38 x^{2} + 13 - 27 x^{2} + 0 = 4$

Step 1.3.2

Add $38 x^{2} + 13 - 27 x^{2}$ and $0$ .

$38 x^{2} + 13 - 27 x^{2} = 4$

Step 1.4

Subtract $27 x^{2}$ from $38 x^{2}$ .

$11 x^{2} + 13 = 4$

Step 2

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

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Step 2.1

Subtract $13$ from both sides of the equation.

$11 x^{2} = 4 - 13$

Step 2.2

Subtract $13$ from $4$ .

$11 x^{2} = - 9$

Step 3

Divide each term in $11 x^{2} = - 9$ by $11$ and simplify.

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Step 3.1

Divide each term in $11 x^{2} = - 9$ by $11$ .

$\frac{11 x^{2}}{11} = \frac{- 9}{11}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $11$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{11 x^{2}}{11} = \frac{- 9}{11}$

Step 3.2.1.2

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

$x^{2} = \frac{- 9}{11}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{9}{11}$

Step 4

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

$x = \pm \sqrt{- \frac{9}{11}}$

Step 5

Simplify $\pm \sqrt{- \frac{9}{11}}$ .

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Step 5.1

Rewrite $- \frac{9}{11}$ as $i^{2} \frac{3^{2}}{11}$ .

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Step 5.1.1

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

$x = \pm \sqrt{i^{2} \frac{9}{11}}$

Step 5.1.2

Rewrite $9$ as $3^{2}$ .

$x = \pm \sqrt{i^{2} \frac{3^{2}}{11}}$

Step 5.2

Pull terms out from under the radical.

$x = \pm i \sqrt{\frac{3^{2}}{11}}$

Step 5.3

Raise $3$ to the power of $2$ .

$x = \pm i \sqrt{\frac{9}{11}}$

Step 5.4

Rewrite $\sqrt{\frac{9}{11}}$ as $\frac{\sqrt{9}}{\sqrt{11}}$ .

$x = \pm i \frac{\sqrt{9}}{\sqrt{11}}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite $9$ as $3^{2}$ .

$x = \pm i \frac{\sqrt{3^{2}}}{\sqrt{11}}$

Step 5.5.2

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

$x = \pm i \frac{3}{\sqrt{11}}$

Step 5.6

Multiply $\frac{3}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm i (\frac{3}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

Step 5.7

Combine and simplify the denominator.

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Step 5.7.1

Multiply $\frac{3}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$x = \pm i \frac{3 \sqrt{11}}{\sqrt{11} \sqrt{11}}$

Step 5.7.2

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm i \frac{3 \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}}$

Step 5.7.3

Raise $\sqrt{11}$ to the power of $1$ .

$x = \pm i \frac{3 \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}}$

Step 5.7.4

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

$x = \pm i \frac{3 \sqrt{11}}{{\sqrt{11}}^{1 + 1}}$

Step 5.7.5

Add $1$ and $1$ .

$x = \pm i \frac{3 \sqrt{11}}{{\sqrt{11}}^{2}}$

Step 5.7.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 5.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$x = \pm i \frac{3 \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}}$

Step 5.7.6.2

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

$x = \pm i \frac{3 \sqrt{11}}{11^{\frac{1}{2} \cdot 2}}$

Step 5.7.6.3

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

$x = \pm i \frac{3 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 5.7.6.4

Cancel the common factor of $2$ .

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Step 5.7.6.4.1

Cancel the common factor.

$x = \pm i \frac{3 \sqrt{11}}{11^{\frac{2}{2}}}$

Step 5.7.6.4.2

Rewrite the expression.

$x = \pm i \frac{3 \sqrt{11}}{11^{1}}$

Step 5.7.6.5

Evaluate the exponent.

$x = \pm i \frac{3 \sqrt{11}}{11}$

Step 5.8

Combine $i$ and $\frac{3 \sqrt{11}}{11}$ .

$x = \pm \frac{i (3 \sqrt{11})}{11}$

Step 5.9

Move $3$ to the left of $i$ .

$x = \pm \frac{3 i \sqrt{11}}{11}$

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.

$x = \frac{3 i \sqrt{11}}{11}$

Step 6.2

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

$x = - \frac{3 i \sqrt{11}}{11}$

Step 6.3

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

$x = \frac{3 i \sqrt{11}}{11}, - \frac{3 i \sqrt{11}}{11}$