Algebra Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

Simplify each term.

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Step 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 2.1.2

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

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Step 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 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 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 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 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 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 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 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 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 2.1.3.1.6

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

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Step 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 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 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 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 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 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 2.1.4

Simplify each term.

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Step 2.1.4.1

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

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Step 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 2.1.4.1.2

Multiply $2$ by $- 2$ .

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

Step 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 2.2

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

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Step 2.2.1

Combine the numerators over the common denominator.

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

Step 2.2.2

Subtract $4$ from $4$ .

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

Step 2.2.3

Divide $0$ by $9$ .

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

Step 2.2.4

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

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

Step 3

Factor $y$ out of $y^{2} - \frac{4 y}{3}$ .

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Step 3.1

Factor $y$ out of $y^{2}$ .

$y \cdot y - \frac{4 y}{3} = 0$

Step 3.2

Factor $y$ out of $- \frac{4 y}{3}$ .

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

Step 3.3

Factor $y$ out of $y \cdot y + y (- \frac{4}{3})$ .

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

Step 4

To write $y$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

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

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

Step 8

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

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

Step 9

Set $y$ equal to $0$ .

$y = 0$

Step 10

Set $\frac{3 y - 4}{3}$ equal to $0$ and solve for $y$ .

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Step 10.1

Set $\frac{3 y - 4}{3}$ equal to $0$ .

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

Step 10.2

Solve $\frac{3 y - 4}{3} = 0$ for $y$ .

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Step 10.2.1

Set the numerator equal to zero.

$3 y - 4 = 0$

Step 10.2.2

Solve the equation for $y$ .

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Step 10.2.2.1

Add $4$ to both sides of the equation.

$3 y = 4$

Step 10.2.2.2

Divide each term in $3 y = 4$ by $3$ and simplify.

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Step 10.2.2.2.1

Divide each term in $3 y = 4$ by $3$ .

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

Step 10.2.2.2.2

Simplify the left side.

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Step 10.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 10.2.2.2.2.1.1

Cancel the common factor.

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

Step 10.2.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 11

The final solution is all the values that make $y (\frac{3 y - 4}{3}) = 0$ true.

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