Basic Math Examples

$(3 y + 9) (3 y + 8) (3 y) = (3 y) (3 y) (4 y + 3)$

Step 1

Simplify $(3 y + 9) (3 y + 8) (3 y)$ .

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Step 1.1

Rewrite.

$0 + 0 + (3 y + 9) (3 y + 8) (3 y) = 3 y (3 y) (4 y + 3)$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Add $0$ and $0$ .

$0 + (3 y + 9) (3 y + 8) (3 y) = 3 y (3 y) (4 y + 3)$

Step 1.2.2

Rewrite using the commutative property of multiplication.

$3 (3 y + 9) (3 y + 8) y = 3 y (3 y) (4 y + 3)$

Step 1.2.3

Apply the distributive property.

$(3 (3 y) + 3 \cdot 9) (3 y + 8) y = 3 y (3 y) (4 y + 3)$

Step 1.2.4

Multiply.

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Step 1.2.4.1

Multiply $3$ by $3$ .

$(9 y + 3 \cdot 9) (3 y + 8) y = 3 y (3 y) (4 y + 3)$

Step 1.2.4.2

Multiply $3$ by $9$ .

$(9 y + 27) (3 y + 8) y = 3 y (3 y) (4 y + 3)$

Step 1.3

Expand $(9 y + 27) (3 y + 8)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$(9 y (3 y + 8) + 27 (3 y + 8)) y = 3 y (3 y) (4 y + 3)$

Step 1.3.2

Apply the distributive property.

$(9 y (3 y) + 9 y \cdot 8 + 27 (3 y + 8)) y = 3 y (3 y) (4 y + 3)$

Step 1.3.3

Apply the distributive property.

$(9 y (3 y) + 9 y \cdot 8 + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$(9 \cdot 3 y \cdot y + 9 y \cdot 8 + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.2

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

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Step 1.4.1.2.1

Move $y$ .

$(9 \cdot 3 (y \cdot y) + 9 y \cdot 8 + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.2.2

Multiply $y$ by $y$ .

$(9 \cdot 3 y^{2} + 9 y \cdot 8 + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.3

Multiply $9$ by $3$ .

$(27 y^{2} + 9 y \cdot 8 + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.4

Multiply $8$ by $9$ .

$(27 y^{2} + 72 y + 27 (3 y) + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.5

Multiply $3$ by $27$ .

$(27 y^{2} + 72 y + 81 y + 27 \cdot 8) y = 3 y (3 y) (4 y + 3)$

Step 1.4.1.6

Multiply $27$ by $8$ .

$(27 y^{2} + 72 y + 81 y + 216) y = 3 y (3 y) (4 y + 3)$

Step 1.4.2

Add $72 y$ and $81 y$ .

$(27 y^{2} + 153 y + 216) y = 3 y (3 y) (4 y + 3)$

Step 1.5

Apply the distributive property.

$27 y^{2} y + 153 y \cdot y + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 1.6.1.1

Move $y$ .

$27 (y \cdot y^{2}) + 153 y \cdot y + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6.1.2

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

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Step 1.6.1.2.1

Raise $y$ to the power of $1$ .

$27 (y^{1} y^{2}) + 153 y \cdot y + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 y^{1 + 2} + 153 y \cdot y + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6.1.3

Add $1$ and $2$ .

$27 y^{3} + 153 y \cdot y + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6.2

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

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Step 1.6.2.1

Move $y$ .

$27 y^{3} + 153 (y \cdot y) + 216 y = 3 y (3 y) (4 y + 3)$

Step 1.6.2.2

Multiply $y$ by $y$ .

$27 y^{3} + 153 y^{2} + 216 y = 3 y (3 y) (4 y + 3)$

Step 2

Simplify $3 y (3 y) (4 y + 3)$ .

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Step 2.1

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

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Step 2.1.1

Move $y$ .

$27 y^{3} + 153 y^{2} + 216 y = 3 (y \cdot y) \cdot 3 (4 y + 3)$

Step 2.1.2

Multiply $y$ by $y$ .

$27 y^{3} + 153 y^{2} + 216 y = 3 y^{2} \cdot 3 (4 y + 3)$

Step 2.2

Simplify by multiplying through.

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Step 2.2.1

Multiply $3$ by $3$ .

$27 y^{3} + 153 y^{2} + 216 y = 9 y^{2} (4 y + 3)$

Step 2.2.2

Apply the distributive property.

$27 y^{3} + 153 y^{2} + 216 y = 9 y^{2} (4 y) + 9 y^{2} \cdot 3$

Step 2.2.3

Simplify the expression.

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Step 2.2.3.1

Rewrite using the commutative property of multiplication.

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 y^{2} y + 9 y^{2} \cdot 3$

Step 2.2.3.2

Multiply $3$ by $9$ .

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 y^{2} y + 27 y^{2}$

Step 2.3

Simplify each term.

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Step 2.3.1

Multiply $y^{2}$ by $y$ by adding the exponents.

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Step 2.3.1.1

Move $y$ .

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 (y \cdot y^{2}) + 27 y^{2}$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Raise $y$ to the power of $1$ .

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 (y^{1} y^{2}) + 27 y^{2}$

Step 2.3.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 y^{1 + 2} + 27 y^{2}$

Step 2.3.1.3

Add $1$ and $2$ .

$27 y^{3} + 153 y^{2} + 216 y = 9 \cdot 4 y^{3} + 27 y^{2}$

Step 2.3.2

Multiply $9$ by $4$ .

$27 y^{3} + 153 y^{2} + 216 y = 36 y^{3} + 27 y^{2}$

Step 3

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

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Step 3.1

Subtract $36 y^{3}$ from both sides of the equation.

$27 y^{3} + 153 y^{2} + 216 y - 36 y^{3} = 27 y^{2}$

Step 3.2

Subtract $27 y^{2}$ from both sides of the equation.

$27 y^{3} + 153 y^{2} + 216 y - 36 y^{3} - 27 y^{2} = 0$

Step 3.3

Subtract $36 y^{3}$ from $27 y^{3}$ .

$- 9 y^{3} + 153 y^{2} + 216 y - 27 y^{2} = 0$

Step 3.4

Subtract $27 y^{2}$ from $153 y^{2}$ .

$- 9 y^{3} + 126 y^{2} + 216 y = 0$

Step 4

Factor $- 9 y$ out of $- 9 y^{3} + 126 y^{2} + 216 y$ .

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Step 4.1

Factor $- 9 y$ out of $- 9 y^{3}$ .

$- 9 y \cdot y^{2} + 126 y^{2} + 216 y = 0$

Step 4.2

Factor $- 9 y$ out of $126 y^{2}$ .

$- 9 y \cdot y^{2} - 9 y (- 14 y) + 216 y = 0$

Step 4.3

Factor $- 9 y$ out of $216 y$ .

$- 9 y \cdot y^{2} - 9 y (- 14 y) - 9 y \cdot - 24 = 0$

Step 4.4

Factor $- 9 y$ out of $- 9 y (y^{2}) - 9 y (- 14 y)$ .

$- 9 y (y^{2} - 14 y) - 9 y \cdot - 24 = 0$

Step 4.5

Factor $- 9 y$ out of $- 9 y (y^{2} - 14 y) - 9 y (- 24)$ .

$- 9 y (y^{2} - 14 y - 24) = 0$

Step 5

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

$y = 0$

$y^{2} - 14 y - 24 = 0$

Step 6

Set $y$ equal to $0$ .

$y = 0$

Step 7

Set $y^{2} - 14 y - 24$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $y^{2} - 14 y - 24$ equal to $0$ .

$y^{2} - 14 y - 24 = 0$

Step 7.2

Solve $y^{2} - 14 y - 24 = 0$ for $y$ .

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Step 7.2.1

Use the quadratic formula to find the solutions.

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

Step 7.2.2

Substitute the values $a = 1$ , $b = - 14$ , and $c = - 24$ into the quadratic formula and solve for $y$ .

$\frac{14 \pm \sqrt{{(- 14)}^{2} - 4 \cdot (1 \cdot - 24)}}{2 \cdot 1}$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Simplify the numerator.

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Step 7.2.3.1.1

Raise $- 14$ to the power of $2$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot 1 \cdot - 24}}{2 \cdot 1}$

Step 7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 24$ .

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Step 7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{14 \pm \sqrt{196 - 4 \cdot - 24}}{2 \cdot 1}$

Step 7.2.3.1.2.2

Multiply $- 4$ by $- 24$ .

$y = \frac{14 \pm \sqrt{196 + 96}}{2 \cdot 1}$

Step 7.2.3.1.3

Add $196$ and $96$ .

$y = \frac{14 \pm \sqrt{292}}{2 \cdot 1}$

Step 7.2.3.1.4

Rewrite $292$ as $2^{2} \cdot 73$ .

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Step 7.2.3.1.4.1

Factor $4$ out of $292$ .

$y = \frac{14 \pm \sqrt{4 (73)}}{2 \cdot 1}$

Step 7.2.3.1.4.2

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

$y = \frac{14 \pm \sqrt{2^{2} \cdot 73}}{2 \cdot 1}$

Step 7.2.3.1.5

Pull terms out from under the radical.

$y = \frac{14 \pm 2 \sqrt{73}}{2 \cdot 1}$

Step 7.2.3.2

Multiply $2$ by $1$ .

$y = \frac{14 \pm 2 \sqrt{73}}{2}$

Step 7.2.3.3

Simplify $\frac{14 \pm 2 \sqrt{73}}{2}$ .

$y = 7 \pm \sqrt{73}$

Step 7.2.4

The final answer is the combination of both solutions.

$y = 7 + \sqrt{73}, 7 - \sqrt{73}$

Step 8

The final solution is all the values that make $- 9 y (y^{2} - 14 y - 24) = 0$ true.

$y = 0, 7 + \sqrt{73}, 7 - \sqrt{73}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$y = 0, 7 + \sqrt{73}, 7 - \sqrt{73}$

Decimal Form:

$y = 0, 15.54400374 \dots, - 1.54400374 \dots$