Basic Math Examples

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

Step 1

Multiply both sides by $2$ .

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

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Move $2$ to the left of ${(3 y - 1)}^{2}$ .

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

$2 {(3 y - 1)}^{2} = y$

Step 3

Solve for $y$ .

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Step 3.1

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

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Step 3.1.1

Subtract $y$ from both sides of the equation.

$2 {(3 y - 1)}^{2} - y = 0$

Step 3.1.2

Simplify each term.

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Step 3.1.2.1

Rewrite ${(3 y - 1)}^{2}$ as $(3 y - 1) (3 y - 1)$ .

$2 ((3 y - 1) (3 y - 1)) - y = 0$

Step 3.1.2.2

Expand $(3 y - 1) (3 y - 1)$ using the FOIL Method.

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Step 3.1.2.2.1

Apply the distributive property.

$2 (3 y (3 y - 1) - 1 (3 y - 1)) - y = 0$

Step 3.1.2.2.2

Apply the distributive property.

$2 (3 y (3 y) + 3 y \cdot - 1 - 1 (3 y - 1)) - y = 0$

Step 3.1.2.2.3

Apply the distributive property.

$2 (3 y (3 y) + 3 y \cdot - 1 - 1 (3 y) - 1 \cdot - 1) - y = 0$

Step 3.1.2.3

Simplify and combine like terms.

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Step 3.1.2.3.1

Simplify each term.

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Step 3.1.2.3.1.1

Rewrite using the commutative property of multiplication.

$2 (3 \cdot 3 y \cdot y + 3 y \cdot - 1 - 1 (3 y) - 1 \cdot - 1) - y = 0$

Step 3.1.2.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 3.1.2.3.1.2.1

Move $y$ .

$2 (3 \cdot 3 (y \cdot y) + 3 y \cdot - 1 - 1 (3 y) - 1 \cdot - 1) - y = 0$

Step 3.1.2.3.1.2.2

Multiply $y$ by $y$ .

$2 (3 \cdot 3 y^{2} + 3 y \cdot - 1 - 1 (3 y) - 1 \cdot - 1) - y = 0$

Step 3.1.2.3.1.3

Multiply $3$ by $3$ .

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

Step 3.1.2.3.1.4

Multiply $- 1$ by $3$ .

$2 (9 y^{2} - 3 y - 1 (3 y) - 1 \cdot - 1) - y = 0$

Step 3.1.2.3.1.5

Multiply $3$ by $- 1$ .

$2 (9 y^{2} - 3 y - 3 y - 1 \cdot - 1) - y = 0$

Step 3.1.2.3.1.6

Multiply $- 1$ by $- 1$ .

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

Step 3.1.2.3.2

Subtract $3 y$ from $- 3 y$ .

$2 (9 y^{2} - 6 y + 1) - y = 0$

Step 3.1.2.4

Apply the distributive property.

$2 (9 y^{2}) + 2 (- 6 y) + 2 \cdot 1 - y = 0$

Step 3.1.2.5

Simplify.

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Step 3.1.2.5.1

Multiply $9$ by $2$ .

$18 y^{2} + 2 (- 6 y) + 2 \cdot 1 - y = 0$

Step 3.1.2.5.2

Multiply $- 6$ by $2$ .

$18 y^{2} - 12 y + 2 \cdot 1 - y = 0$

Step 3.1.2.5.3

Multiply $2$ by $1$ .

$18 y^{2} - 12 y + 2 - y = 0$

Step 3.1.3

Subtract $y$ from $- 12 y$ .

$18 y^{2} - 13 y + 2 = 0$

Step 3.2

Factor by grouping.

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Step 3.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 18 \cdot 2 = 36$ and whose sum is $b = - 13$ .

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Step 3.2.1.1

Factor $- 13$ out of $- 13 y$ .

$18 y^{2} - 13 y + 2 = 0$

Step 3.2.1.2

Rewrite $- 13$ as $- 4$ plus $- 9$

$18 y^{2} + (- 4 - 9) y + 2 = 0$

Step 3.2.1.3

Apply the distributive property.

$18 y^{2} - 4 y - 9 y + 2 = 0$

Step 3.2.2

Factor out the greatest common factor from each group.

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Step 3.2.2.1

Group the first two terms and the last two terms.

$(18 y^{2} - 4 y) - 9 y + 2 = 0$

Step 3.2.2.2

Factor out the greatest common factor (GCF) from each group.

$2 y (9 y - 2) - (9 y - 2) = 0$

Step 3.2.3

Factor the polynomial by factoring out the greatest common factor, $9 y - 2$ .

$(9 y - 2) (2 y - 1) = 0$

Step 3.3

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

$9 y - 2 = 0$

$2 y - 1 = 0$

Step 3.4

Set $9 y - 2$ equal to $0$ and solve for $y$ .

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Step 3.4.1

Set $9 y - 2$ equal to $0$ .

$9 y - 2 = 0$

Step 3.4.2

Solve $9 y - 2 = 0$ for $y$ .

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Step 3.4.2.1

Add $2$ to both sides of the equation.

$9 y = 2$

Step 3.4.2.2

Divide each term in $9 y = 2$ by $9$ and simplify.

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Step 3.4.2.2.1

Divide each term in $9 y = 2$ by $9$ .

$\frac{9 y}{9} = \frac{2}{9}$

Step 3.4.2.2.2

Simplify the left side.

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Step 3.4.2.2.2.1

Cancel the common factor of $9$ .

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Step 3.4.2.2.2.1.1

Cancel the common factor.

$\frac{9 y}{9} = \frac{2}{9}$

Step 3.4.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2}{9}$

Step 3.5

Set $2 y - 1$ equal to $0$ and solve for $y$ .

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Step 3.5.1

Set $2 y - 1$ equal to $0$ .

$2 y - 1 = 0$

Step 3.5.2

Solve $2 y - 1 = 0$ for $y$ .

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Step 3.5.2.1

Add $1$ to both sides of the equation.

$2 y = 1$

Step 3.5.2.2

Divide each term in $2 y = 1$ by $2$ and simplify.

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Step 3.5.2.2.1

Divide each term in $2 y = 1$ by $2$ .

$\frac{2 y}{2} = \frac{1}{2}$

Step 3.5.2.2.2

Simplify the left side.

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Step 3.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 3.5.2.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{1}{2}$

Step 3.5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{2}$

Step 3.6

The final solution is all the values that make $(9 y - 2) (2 y - 1) = 0$ true.

$y = \frac{2}{9}, \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$y = \frac{2}{9}, \frac{1}{2}$

Decimal Form:

$y = 0.\overline{2}, 0.5$