Algebra Examples

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

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Subtract $\frac{4}{9}$ from both sides of the equation.

${(y - \frac{2}{3})}^{2} - \frac{4}{9} = 0$

Step 1.2

Simplify ${(y - \frac{2}{3})}^{2} - \frac{4}{9}$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(y - \frac{2}{3})}^{2}$ as $(y - \frac{2}{3}) (y - \frac{2}{3})$ .

$(y - \frac{2}{3}) (y - \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.2

Expand $(y - \frac{2}{3}) (y - \frac{2}{3})$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$y (y - \frac{2}{3}) - \frac{2}{3} (y - \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.2.2

Apply the distributive property.

$y \cdot y + y (- \frac{2}{3}) - \frac{2}{3} (y - \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.2.3

Apply the distributive property.

$y \cdot y + y (- \frac{2}{3}) - \frac{2}{3} y - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y (- \frac{2}{3}) - \frac{2}{3} y - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3.1.2

Combine $y$ and $\frac{2}{3}$ .

$y^{2} - \frac{y \cdot 2}{3} - \frac{2}{3} y - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3.1.3

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

$y^{2} - \frac{2 \cdot y}{3} - \frac{2}{3} y - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3.1.4

Combine $y$ and $\frac{2}{3}$ .

$y^{2} - \frac{2 y}{3} - \frac{y \cdot 2}{3} - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3.1.5

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

$y^{2} - \frac{2 y}{3} - \frac{2 \cdot y}{3} - \frac{2}{3} (- \frac{2}{3}) - \frac{4}{9} = 0$

Step 1.2.1.3.1.6

Multiply $- \frac{2}{3} (- \frac{2}{3})$ .

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Step 1.2.1.3.1.6.1

Multiply $- 1$ by $- 1$ .

$y^{2} - \frac{2 y}{3} - \frac{2 y}{3} + 1 (\frac{2}{3}) \frac{2}{3} - \frac{4}{9} = 0$

Step 1.2.1.3.1.6.2

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

$y^{2} - \frac{2 y}{3} - \frac{2 y}{3} + \frac{2}{3} \cdot \frac{2}{3} - \frac{4}{9} = 0$

Step 1.2.1.3.1.6.3

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

$y^{2} - \frac{2 y}{3} - \frac{2 y}{3} + \frac{2 \cdot 2}{3 \cdot 3} - \frac{4}{9} = 0$

Step 1.2.1.3.1.6.4

Multiply $2$ by $2$ .

$y^{2} - \frac{2 y}{3} - \frac{2 y}{3} + \frac{4}{3 \cdot 3} - \frac{4}{9} = 0$

Step 1.2.1.3.1.6.5

Multiply $3$ by $3$ .

$y^{2} - \frac{2 y}{3} - \frac{2 y}{3} + \frac{4}{9} - \frac{4}{9} = 0$

Step 1.2.1.3.2

Subtract $\frac{2 y}{3}$ from $- \frac{2 y}{3}$ .

$y^{2} - 2 \frac{2 y}{3} + \frac{4}{9} - \frac{4}{9} = 0$

Step 1.2.1.4

Simplify each term.

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Step 1.2.1.4.1

Multiply $- 2 \frac{2 y}{3}$ .

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Step 1.2.1.4.1.1

Combine $- 2$ and $\frac{2 y}{3}$ .

$y^{2} + \frac{- 2 (2 y)}{3} + \frac{4}{9} - \frac{4}{9} = 0$

Step 1.2.1.4.1.2

Multiply $2$ by $- 2$ .

$y^{2} + \frac{- 4 y}{3} + \frac{4}{9} - \frac{4}{9} = 0$

Step 1.2.1.4.2

Move the negative in front of the fraction.

$y^{2} - \frac{4 y}{3} + \frac{4}{9} - \frac{4}{9} = 0$

Step 1.2.2

Combine the opposite terms in $y^{2} - \frac{4 y}{3} + \frac{4}{9} - \frac{4}{9}$ .

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Step 1.2.2.1

Combine the numerators over the common denominator.

$y^{2} - \frac{4 y}{3} + \frac{4 - 4}{9} = 0$

Step 1.2.2.2

Subtract $4$ from $4$ .

$y^{2} - \frac{4 y}{3} + \frac{0}{9} = 0$

Step 1.2.2.3

Divide $0$ by $9$ .

$y^{2} - \frac{4 y}{3} + 0 = 0$

Step 1.2.2.4

Add $y^{2} - \frac{4 y}{3}$ and $0$ .

$y^{2} - \frac{4 y}{3} = 0$

Step 2

Multiply through by the least common denominator $3$ , then simplify.

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Step 2.1

Apply the distributive property.

$3 y^{2} + 3 (- \frac{4 y}{3}) = 0$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Move the leading negative in $- \frac{4 y}{3}$ into the numerator.

$3 y^{2} + 3 (\frac{- 4 y}{3}) = 0$

Step 2.2.2

Cancel the common factor.

$3 y^{2} + 3 (\frac{- 4 y}{3}) = 0$

Step 2.2.3

Rewrite the expression.

$3 y^{2} - 4 y = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 3$ , $b = - 4$ , and $c = 0$ into the quadratic formula and solve for $y$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (3 \cdot 0)}}{2 \cdot 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 4$ to the power of $2$ .

$y = \frac{4 \pm \sqrt{16 - 4 \cdot 3 \cdot 0}}{2 \cdot 3}$

Step 5.1.2

Multiply $- 4 \cdot 3 \cdot 0$ .

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Step 5.1.2.1

Multiply $- 4$ by $3$ .

$y = \frac{4 \pm \sqrt{16 - 12 \cdot 0}}{2 \cdot 3}$

Step 5.1.2.2

Multiply $- 12$ by $0$ .

$y = \frac{4 \pm \sqrt{16 + 0}}{2 \cdot 3}$

Step 5.1.3

Add $16$ and $0$ .

$y = \frac{4 \pm \sqrt{16}}{2 \cdot 3}$

Step 5.1.4

Rewrite $16$ as $4^{2}$ .

$y = \frac{4 \pm \sqrt{4^{2}}}{2 \cdot 3}$

Step 5.1.5

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

$y = \frac{4 \pm 4}{2 \cdot 3}$

Step 5.2

Multiply $2$ by $3$ .

$y = \frac{4 \pm 4}{6}$

Step 5.3

Simplify $\frac{4 \pm 4}{6}$ .

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

Step 6

The final answer is the combination of both solutions.

$y = \frac{4}{3}, 0$