Solve by Completing the Square 9y^2-12y-2=0

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

Step 1

Add $2$ to both sides of the equation.

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

Step 2

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

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

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $9$ .

Cancel the common factor.

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

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

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

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

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

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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.

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

Step 5

Simplify the equation.

Simplify the left side.

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify the right side.

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

Simplify each term.

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

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

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

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

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

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

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

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

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

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

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

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

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

Simplify terms.

Combine the numerators over the common denominator.

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

Add $2$ and $4$ .

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

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

Factor $3$ out of $6$ .

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

Cancel the common factors.

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

Step 7

Solve the equation for $y$ .

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

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

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

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

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

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

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

Solve by Completing the Square 9y^2-12y-2=0

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