Algebra Examples

$9 x^{2} - 12 x - 2 = 0$

Step 1

Add $2$ to both sides of the equation.

$9 x^{2} - 12 x = 2$

Step 2

Divide each term in $9 x^{2} - 12 x = 2$ by $9$ and simplify.

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Step 2.1

Divide each term in $9 x^{2} - 12 x = 2$ by $9$ .

$\frac{9 x^{2}}{9} + \frac{- 12 x}{9} = \frac{2}{9}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $9$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{9 x^{2}}{9} + \frac{- 12 x}{9} = \frac{2}{9}$

Step 2.2.1.1.2

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

$x^{2} + \frac{- 12 x}{9} = \frac{2}{9}$

Step 2.2.1.2

Cancel the common factor of $- 12$ and $9$ .

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Step 2.2.1.2.1

Factor $3$ out of $- 12 x$ .

$x^{2} + \frac{3 (- 4 x)}{9} = \frac{2}{9}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $3$ out of $9$ .

$x^{2} + \frac{3 (- 4 x)}{3 (3)} = \frac{2}{9}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{3 (- 4 x)}{3 \cdot 3} = \frac{2}{9}$

Step 2.2.1.2.2.3

Rewrite the expression.

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

Step 2.2.1.3

Move the negative in front of the fraction.

$x^{2} - \frac{4 x}{3} = \frac{2}{9}$

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

${(\frac{b}{2})}^{2} = {(- \frac{2}{3})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{4 x}{3} + {(- \frac{2}{3})}^{2} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

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

$x^{2} - \frac{4 x}{3} + {(- 1)}^{2} {(\frac{2}{3})}^{2} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.1.1.1.2

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

$x^{2} - \frac{4 x}{3} + {(- 1)}^{2} \frac{2^{2}}{3^{2}} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{4 x}{3} + 1 \frac{2^{2}}{3^{2}} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.1.1.3

Multiply $\frac{2^{2}}{3^{2}}$ by $1$ .

$x^{2} - \frac{4 x}{3} + \frac{2^{2}}{3^{2}} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.1.1.4

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

$x^{2} - \frac{4 x}{3} + \frac{4}{3^{2}} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.1.1.5

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

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{2}{9} + {(- \frac{2}{3})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{2}{9} + {(- \frac{2}{3})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

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

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{2}{9} + {(- 1)}^{2} {(\frac{2}{3})}^{2}$

Step 5.2.1.1.1.2

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

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{2}{9} + {(- 1)}^{2} \frac{2^{2}}{3^{2}}$

Step 5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{2}{9} + 1 \frac{2^{2}}{3^{2}}$

Step 5.2.1.1.3

Multiply $\frac{2^{2}}{3^{2}}$ by $1$ .

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

Step 5.2.1.1.4

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

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

Step 5.2.1.1.5

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

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

Step 5.2.1.2

Simplify terms.

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Step 5.2.1.2.1

Combine the numerators over the common denominator.

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

Step 5.2.1.2.2

Add $2$ and $4$ .

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{6}{9}$

Step 5.2.1.2.3

Cancel the common factor of $6$ and $9$ .

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Step 5.2.1.2.3.1

Factor $3$ out of $6$ .

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{3 (2)}{9}$

Step 5.2.1.2.3.2

Cancel the common factors.

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Step 5.2.1.2.3.2.1

Factor $3$ out of $9$ .

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{3 \cdot 2}{3 \cdot 3}$

Step 5.2.1.2.3.2.2

Cancel the common factor.

$x^{2} - \frac{4 x}{3} + \frac{4}{9} = \frac{3 \cdot 2}{3 \cdot 3}$

Step 5.2.1.2.3.2.3

Rewrite the expression.

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

Step 6

Factor the perfect trinomial square into ${(x - \frac{2}{3})}^{2}$ .

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

Step 7

Solve the equation for $x$ .

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Step 7.1

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

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

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Combine and simplify the denominator.

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Step 7.2.3.1

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

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

Step 7.2.3.2

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

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

Step 7.2.3.3

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

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

Step 7.2.3.4

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

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

Step 7.2.3.5

Add $1$ and $1$ .

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

Step 7.2.3.6

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

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Step 7.2.3.6.1

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

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

Step 7.2.3.6.2

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

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

Step 7.2.3.6.3

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

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

Step 7.2.3.6.4

Cancel the common factor of $2$ .

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Step 7.2.3.6.4.1

Cancel the common factor.

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

Step 7.2.3.6.4.2

Rewrite the expression.

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

Step 7.2.3.6.5

Evaluate the exponent.

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

Step 7.2.4

Simplify the numerator.

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Step 7.2.4.1

Combine using the product rule for radicals.

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

Step 7.2.4.2

Multiply $2$ by $3$ .

$x - \frac{2}{3} = \pm \frac{\sqrt{6}}{3}$

Step 7.3

Add $\frac{2}{3}$ to both sides of the equation.

$x = \pm \frac{\sqrt{6}}{3} + \frac{2}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm \frac{\sqrt{6}}{3} + \frac{2}{3}$

Decimal Form:

$x = 1.48316324 \dots, - 0.14982991 \dots$