Algebra Examples

${(3 y + 9)}^{2} = 81$

Step 1

Subtract $81$ from both sides of the equation.

${(3 y + 9)}^{2} - 81 = 0$

Step 2

Simplify ${(3 y + 9)}^{2} - 81$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(3 y + 9)}^{2}$ as $(3 y + 9) (3 y + 9)$ .

$(3 y + 9) (3 y + 9) - 81 = 0$

Step 2.1.2

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

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Step 2.1.2.1

Apply the distributive property.

$3 y (3 y + 9) + 9 (3 y + 9) - 81 = 0$

Step 2.1.2.2

Apply the distributive property.

$3 y (3 y) + 3 y \cdot 9 + 9 (3 y + 9) - 81 = 0$

Step 2.1.2.3

Apply the distributive property.

$3 y (3 y) + 3 y \cdot 9 + 9 (3 y) + 9 \cdot 9 - 81 = 0$

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

Rewrite using the commutative property of multiplication.

$3 \cdot 3 y \cdot y + 3 y \cdot 9 + 9 (3 y) + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $y$ .

$3 \cdot 3 (y \cdot y) + 3 y \cdot 9 + 9 (3 y) + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.2.2

Multiply $y$ by $y$ .

$3 \cdot 3 y^{2} + 3 y \cdot 9 + 9 (3 y) + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.3

Multiply $3$ by $3$ .

$9 y^{2} + 3 y \cdot 9 + 9 (3 y) + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.4

Multiply $9$ by $3$ .

$9 y^{2} + 27 y + 9 (3 y) + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.5

Multiply $3$ by $9$ .

$9 y^{2} + 27 y + 27 y + 9 \cdot 9 - 81 = 0$

Step 2.1.3.1.6

Multiply $9$ by $9$ .

$9 y^{2} + 27 y + 27 y + 81 - 81 = 0$

Step 2.1.3.2

Add $27 y$ and $27 y$ .

$9 y^{2} + 54 y + 81 - 81 = 0$

Step 2.2

Combine the opposite terms in $9 y^{2} + 54 y + 81 - 81$ .

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Step 2.2.1

Subtract $81$ from $81$ .

$9 y^{2} + 54 y + 0 = 0$

Step 2.2.2

Add $9 y^{2} + 54 y$ and $0$ .

$9 y^{2} + 54 y = 0$

Step 3

Factor $9 y$ out of $9 y^{2} + 54 y$ .

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Step 3.1

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

$9 y (y) + 54 y = 0$

Step 3.2

Factor $9 y$ out of $54 y$ .

$9 y (y) + 9 y (6) = 0$

Step 3.3

Factor $9 y$ out of $9 y (y) + 9 y (6)$ .

$9 y (y + 6) = 0$

Step 4

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

$y = 0$

$y + 6 = 0$

Step 5

Set $y$ equal to $0$ .

$y = 0$

Step 6

Set $y + 6$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y + 6$ equal to $0$ .

$y + 6 = 0$

Step 6.2

Subtract $6$ from both sides of the equation.

$y = - 6$

Step 7

The final solution is all the values that make $9 y (y + 6) = 0$ true.

$y = 0, - 6$