Basic Math Examples

${(\frac{- 2 y + 12}{3})}^{2} - y + 2 y^{2} = 12$

Step 1

Simplify ${(\frac{- 2 y + 12}{3})}^{2} - y + 2 y^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Factor $2$ out of $- 2 y + 12$ .

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Step 1.1.1.1

Factor $2$ out of $- 2 y$ .

${(\frac{2 (- y) + 12}{3})}^{2} - y + 2 y^{2} = 12$

Step 1.1.1.2

Factor $2$ out of $12$ .

${(\frac{2 (- y) + 2 (6)}{3})}^{2} - y + 2 y^{2} = 12$

Step 1.1.1.3

Factor $2$ out of $2 (- y) + 2 (6)$ .

${(\frac{2 (- y + 6)}{3})}^{2} - y + 2 y^{2} = 12$

Step 1.1.2

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

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Step 1.1.2.1

Apply the product rule to $\frac{2 (- y + 6)}{3}$ .

$\frac{{(2 (- y + 6))}^{2}}{3^{2}} - y + 2 y^{2} = 12$

Step 1.1.2.2

Apply the product rule to $2 (- y + 6)$ .

$\frac{2^{2} {(- y + 6)}^{2}}{3^{2}} - y + 2 y^{2} = 12$

Step 1.1.3

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

$\frac{4 {(- y + 6)}^{2}}{3^{2}} - y + 2 y^{2} = 12$

Step 1.1.4

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

$\frac{4 {(- y + 6)}^{2}}{9} - y + 2 y^{2} = 12$

Step 1.2

To write $- y$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4 {(- y + 6)}^{2}}{9} - y \cdot \frac{9}{9} + 2 y^{2} = 12$

Step 1.3

Combine $- y$ and $\frac{9}{9}$ .

$\frac{4 {(- y + 6)}^{2}}{9} + \frac{- y \cdot 9}{9} + 2 y^{2} = 12$

Step 1.4

Combine the numerators over the common denominator.

$\frac{4 {(- y + 6)}^{2} - y \cdot 9}{9} + 2 y^{2} = 12$

Step 1.5

Multiply $9$ by $- 1$ .

$\frac{4 {(- y + 6)}^{2} - 9 y}{9} + 2 y^{2} = 12$

Step 2

Multiply each term in $\frac{4 {(- y + 6)}^{2} - 9 y}{9} + 2 y^{2} = 12$ by $9$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{4 {(- y + 6)}^{2} - 9 y}{9} + 2 y^{2} = 12$ by $9$ .

$\frac{4 {(- y + 6)}^{2} - 9 y}{9} \cdot 9 + 2 y^{2} \cdot 9 = 12 \cdot 9$

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.

$\frac{4 {(- y + 6)}^{2} - 9 y}{9} \cdot 9 + 2 y^{2} \cdot 9 = 12 \cdot 9$

Step 2.2.1.1.2

Rewrite the expression.

$4 {(- y + 6)}^{2} - 9 y + 2 y^{2} \cdot 9 = 12 \cdot 9$

Step 2.2.1.2

Multiply $9$ by $2$ .

$4 {(- y + 6)}^{2} - 9 y + 18 y^{2} = 12 \cdot 9$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $12$ by $9$ .

$4 {(- y + 6)}^{2} - 9 y + 18 y^{2} = 108$

Step 3

Subtract $108$ from both sides of the equation.

$4 {(- y + 6)}^{2} - 9 y + 18 y^{2} - 108 = 0$

Step 4

Simplify $4 {(- y + 6)}^{2} - 9 y + 18 y^{2} - 108$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Rewrite ${(- y + 6)}^{2}$ as $(- y + 6) (- y + 6)$ .

$4 ((- y + 6) (- y + 6)) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.2

Expand $(- y + 6) (- y + 6)$ using the FOIL Method.

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Step 4.1.2.1

Apply the distributive property.

$4 (- y (- y + 6) + 6 (- y + 6)) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.2.2

Apply the distributive property.

$4 (- y (- y) - y \cdot 6 + 6 (- y + 6)) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.2.3

Apply the distributive property.

$4 (- y (- y) - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3

Simplify and combine like terms.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Rewrite using the commutative property of multiplication.

$4 (- 1 \cdot - 1 y \cdot y - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.2

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

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Step 4.1.3.1.2.1

Move $y$ .

$4 (- 1 \cdot - 1 (y \cdot y) - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.2.2

Multiply $y$ by $y$ .

$4 (- 1 \cdot - 1 y^{2} - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.3

Multiply $- 1$ by $- 1$ .

$4 (1 y^{2} - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.4

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

$4 (y^{2} - y \cdot 6 + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.5

Multiply $6$ by $- 1$ .

$4 (y^{2} - 6 y + 6 (- y) + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.6

Multiply $- 1$ by $6$ .

$4 (y^{2} - 6 y - 6 y + 6 \cdot 6) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.1.7

Multiply $6$ by $6$ .

$4 (y^{2} - 6 y - 6 y + 36) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.3.2

Subtract $6 y$ from $- 6 y$ .

$4 (y^{2} - 12 y + 36) - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.4

Apply the distributive property.

$4 y^{2} + 4 (- 12 y) + 4 \cdot 36 - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.5

Simplify.

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Step 4.1.5.1

Multiply $- 12$ by $4$ .

$4 y^{2} - 48 y + 4 \cdot 36 - 9 y + 18 y^{2} - 108 = 0$

Step 4.1.5.2

Multiply $4$ by $36$ .

$4 y^{2} - 48 y + 144 - 9 y + 18 y^{2} - 108 = 0$

Step 4.2

Add $4 y^{2}$ and $18 y^{2}$ .

$22 y^{2} - 48 y + 144 - 9 y - 108 = 0$

Step 4.3

Subtract $9 y$ from $- 48 y$ .

$22 y^{2} - 57 y + 144 - 108 = 0$

Step 4.4

Subtract $108$ from $144$ .

$22 y^{2} - 57 y + 36 = 0$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 22 \cdot 36 = 792$ and whose sum is $b = - 57$ .

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Step 5.1.1

Factor $- 57$ out of $- 57 y$ .

$22 y^{2} - 57 y + 36 = 0$

Step 5.1.2

Rewrite $- 57$ as $- 24$ plus $- 33$

$22 y^{2} + (- 24 - 33) y + 36 = 0$

Step 5.1.3

Apply the distributive property.

$22 y^{2} - 24 y - 33 y + 36 = 0$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$(22 y^{2} - 24 y) - 33 y + 36 = 0$

Step 5.2.2

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

$2 y (11 y - 12) - 3 (11 y - 12) = 0$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $11 y - 12$ .

$(11 y - 12) (2 y - 3) = 0$

Step 6

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

$11 y - 12 = 0$

$2 y - 3 = 0$

Step 7

Set $11 y - 12$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $11 y - 12$ equal to $0$ .

$11 y - 12 = 0$

Step 7.2

Solve $11 y - 12 = 0$ for $y$ .

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Step 7.2.1

Add $12$ to both sides of the equation.

$11 y = 12$

Step 7.2.2

Divide each term in $11 y = 12$ by $11$ and simplify.

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Step 7.2.2.1

Divide each term in $11 y = 12$ by $11$ .

$\frac{11 y}{11} = \frac{12}{11}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $11$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{11 y}{11} = \frac{12}{11}$

Step 7.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{12}{11}$

Step 8

Set $2 y - 3$ equal to $0$ and solve for $y$ .

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Step 8.1

Set $2 y - 3$ equal to $0$ .

$2 y - 3 = 0$

Step 8.2

Solve $2 y - 3 = 0$ for $y$ .

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Step 8.2.1

Add $3$ to both sides of the equation.

$2 y = 3$

Step 8.2.2

Divide each term in $2 y = 3$ by $2$ and simplify.

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Step 8.2.2.1

Divide each term in $2 y = 3$ by $2$ .

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

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.2.1.2

Divide $y$ by $1$ .

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

Step 9

The final solution is all the values that make $(11 y - 12) (2 y - 3) = 0$ true.

$y = \frac{12}{11}, \frac{3}{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$y = \frac{12}{11}, \frac{3}{2}$

Decimal Form:

$y = 1.\overline{09}, 1.5$

Mixed Number Form:

$y = 1 \frac{1}{11}, 1 \frac{1}{2}$