Basic Math Examples

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

Step 1

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

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Step 1.1

Rewrite ${(y - 9)}^{2}$ as $(y - 9) (y - 9)$ .

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

Step 1.2

Expand $(y - 9) (y - 9)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $y$ by $y$ .

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

Step 1.3.1.2

Move $- 9$ to the left of $y$ .

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

Step 1.3.1.3

Multiply $- 9$ by $- 9$ .

$y = - \frac{2}{3} \cdot (y^{2} - 9 y - 9 y + 81)$

Step 1.3.2

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

$y = - \frac{2}{3} \cdot (y^{2} - 18 y + 81)$

Step 1.4

Apply the distributive property.

$y = - \frac{2}{3} y^{2} - \frac{2}{3} (- 18 y) - \frac{2}{3} \cdot 81$

Step 1.5

Simplify.

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Step 1.5.1

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

$y = - \frac{y^{2} \cdot 2}{3} - \frac{2}{3} (- 18 y) - \frac{2}{3} \cdot 81$

Step 1.5.2

Cancel the common factor of $3$ .

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Step 1.5.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - \frac{y^{2} \cdot 2}{3} + \frac{- 2}{3} (- 18 y) - \frac{2}{3} \cdot 81$

Step 1.5.2.2

Factor $3$ out of $- 18 y$ .

$y = - \frac{y^{2} \cdot 2}{3} + \frac{- 2}{3} (3 (- 6 y)) - \frac{2}{3} \cdot 81$

Step 1.5.2.3

Cancel the common factor.

$y = - \frac{y^{2} \cdot 2}{3} + \frac{- 2}{3} (3 (- 6 y)) - \frac{2}{3} \cdot 81$

Step 1.5.2.4

Rewrite the expression.

$y = - \frac{y^{2} \cdot 2}{3} - 2 (- 6 y) - \frac{2}{3} \cdot 81$

Step 1.5.3

Multiply $- 6$ by $- 2$ .

$y = - \frac{y^{2} \cdot 2}{3} + 12 y - \frac{2}{3} \cdot 81$

Step 1.5.4

Cancel the common factor of $3$ .

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Step 1.5.4.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$y = - \frac{y^{2} \cdot 2}{3} + 12 y + \frac{- 2}{3} \cdot 81$

Step 1.5.4.2

Factor $3$ out of $81$ .

$y = - \frac{y^{2} \cdot 2}{3} + 12 y + \frac{- 2}{3} \cdot (3 (27))$

Step 1.5.4.3

Cancel the common factor.

$y = - \frac{y^{2} \cdot 2}{3} + 12 y + \frac{- 2}{3} \cdot (3 \cdot 27)$

Step 1.5.4.4

Rewrite the expression.

$y = - \frac{y^{2} \cdot 2}{3} + 12 y - 2 \cdot 27$

Step 1.5.5

Multiply $- 2$ by $27$ .

$y = - \frac{y^{2} \cdot 2}{3} + 12 y - 54$

Step 1.6

Move $2$ to the left of $y^{2}$ .

$y = - \frac{2 y^{2}}{3} + 12 y - 54$

Step 2

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- \frac{2 y^{2}}{3} + 12 y - 54 = y$

Step 3

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

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Step 3.1

Subtract $y$ from both sides of the equation.

$- \frac{2 y^{2}}{3} + 12 y - 54 - y = 0$

Step 3.2

Subtract $y$ from $12 y$ .

$- \frac{2 y^{2}}{3} + 11 y - 54 = 0$

Step 4

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

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Step 4.1

Apply the distributive property.

$3 (- \frac{2 y^{2}}{3}) + 3 (11 y) + 3 \cdot - 54 = 0$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Move the leading negative in $- \frac{2 y^{2}}{3}$ into the numerator.

$3 (\frac{- 2 y^{2}}{3}) + 3 (11 y) + 3 \cdot - 54 = 0$

Step 4.2.1.2

Cancel the common factor.

$3 (\frac{- 2 y^{2}}{3}) + 3 (11 y) + 3 \cdot - 54 = 0$

Step 4.2.1.3

Rewrite the expression.

$- 2 y^{2} + 3 (11 y) + 3 \cdot - 54 = 0$

Step 4.2.2

Multiply $11$ by $3$ .

$- 2 y^{2} + 33 y + 3 \cdot - 54 = 0$

Step 4.2.3

Multiply $3$ by $- 54$ .

$- 2 y^{2} + 33 y - 162 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = - 2$ , $b = 33$ , and $c = - 162$ into the quadratic formula and solve for $y$ .

$\frac{- 33 \pm \sqrt{33^{2} - 4 \cdot (- 2 \cdot - 162)}}{2 \cdot - 2}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $33$ to the power of $2$ .

$y = \frac{- 33 \pm \sqrt{1089 - 4 \cdot - 2 \cdot - 162}}{2 \cdot - 2}$

Step 7.1.2

Multiply $- 4 \cdot - 2 \cdot - 162$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 2$ .

$y = \frac{- 33 \pm \sqrt{1089 + 8 \cdot - 162}}{2 \cdot - 2}$

Step 7.1.2.2

Multiply $8$ by $- 162$ .

$y = \frac{- 33 \pm \sqrt{1089 - 1296}}{2 \cdot - 2}$

Step 7.1.3

Subtract $1296$ from $1089$ .

$y = \frac{- 33 \pm \sqrt{- 207}}{2 \cdot - 2}$

Step 7.1.4

Rewrite $- 207$ as $- 1 (207)$ .

$y = \frac{- 33 \pm \sqrt{- 1 \cdot 207}}{2 \cdot - 2}$

Step 7.1.5

Rewrite $\sqrt{- 1 (207)}$ as $\sqrt{- 1} \cdot \sqrt{207}$ .

$y = \frac{- 33 \pm \sqrt{- 1} \cdot \sqrt{207}}{2 \cdot - 2}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 33 \pm i \cdot \sqrt{207}}{2 \cdot - 2}$

Step 7.1.7

Rewrite $207$ as $3^{2} \cdot 23$ .

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Step 7.1.7.1

Factor $9$ out of $207$ .

$y = \frac{- 33 \pm i \cdot \sqrt{9 (23)}}{2 \cdot - 2}$

Step 7.1.7.2

Rewrite $9$ as $3^{2}$ .

$y = \frac{- 33 \pm i \cdot \sqrt{3^{2} \cdot 23}}{2 \cdot - 2}$

Step 7.1.8

Pull terms out from under the radical.

$y = \frac{- 33 \pm i \cdot (3 \sqrt{23})}{2 \cdot - 2}$

Step 7.1.9

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

$y = \frac{- 33 \pm 3 i \sqrt{23}}{2 \cdot - 2}$

Step 7.2

Multiply $2$ by $- 2$ .

$y = \frac{- 33 \pm 3 i \sqrt{23}}{- 4}$

Step 7.3

Simplify $\frac{- 33 \pm 3 i \sqrt{23}}{- 4}$ .

$y = \frac{33 \pm 3 i \sqrt{23}}{4}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{33 + 3 i \sqrt{23}}{4}, \frac{33 - 3 i \sqrt{23}}{4}$