Examples

${(2 x + 9)}^{2} - {(8 y - 6)}^{2} = 4$

Step 1

Simplify the left side ${(2 x + 9)}^{2} - {(8 y - 6)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

$(2 x + 9) (2 x + 9) - {(8 y - 6)}^{2} = 4$

Step 1.1.2

Expand $(2 x + 9) (2 x + 9)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$2 x (2 x + 9) + 9 (2 x + 9) - {(8 y - 6)}^{2} = 4$

Step 1.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 9 + 9 (2 x + 9) - {(8 y - 6)}^{2} = 4$

Step 1.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot (2 x \cdot x) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.3.1.2.1

Move $x$ .

$2 \cdot (2 (x \cdot x)) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot (2 x^{2}) + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot 9 + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.4

Multiply $9$ by $2$ .

$4 x^{2} + 18 x + 9 (2 x) + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.5

Multiply $2$ by $9$ .

$4 x^{2} + 18 x + 18 x + 9 \cdot 9 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.1.6

Multiply $9$ by $9$ .

$4 x^{2} + 18 x + 18 x + 81 - {(8 y - 6)}^{2} = 4$

Step 1.1.3.2

Add $18 x$ and $18 x$ .

$4 x^{2} + 36 x + 81 - {(8 y - 6)}^{2} = 4$

Step 1.1.4

Rewrite ${(8 y - 6)}^{2}$ as $(8 y - 6) (8 y - 6)$ .

$4 x^{2} + 36 x + 81 - ((8 y - 6) (8 y - 6)) = 4$

Step 1.1.5

Expand $(8 y - 6) (8 y - 6)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

$4 x^{2} + 36 x + 81 - (8 y (8 y - 6) - 6 (8 y - 6)) = 4$

Step 1.1.5.2

Apply the distributive property.

$4 x^{2} + 36 x + 81 - (8 y (8 y) + 8 y \cdot - 6 - 6 (8 y - 6)) = 4$

Step 1.1.5.3

Apply the distributive property.

$4 x^{2} + 36 x + 81 - (8 y (8 y) + 8 y \cdot - 6 - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6

Simplify and combine like terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

Rewrite using the commutative property of multiplication.

$4 x^{2} + 36 x + 81 - (8 \cdot (8 y \cdot y) + 8 y \cdot - 6 - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6.1.2

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

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Step 1.1.6.1.2.1

Move $y$ .

$4 x^{2} + 36 x + 81 - (8 \cdot (8 (y \cdot y)) + 8 y \cdot - 6 - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6.1.2.2

Multiply $y$ by $y$ .

$4 x^{2} + 36 x + 81 - (8 \cdot (8 y^{2}) + 8 y \cdot - 6 - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6.1.3

Multiply $8$ by $8$ .

$4 x^{2} + 36 x + 81 - (64 y^{2} + 8 y \cdot - 6 - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6.1.4

Multiply $- 6$ by $8$ .

$4 x^{2} + 36 x + 81 - (64 y^{2} - 48 y - 6 (8 y) - 6 \cdot - 6) = 4$

Step 1.1.6.1.5

Multiply $8$ by $- 6$ .

$4 x^{2} + 36 x + 81 - (64 y^{2} - 48 y - 48 y - 6 \cdot - 6) = 4$

Step 1.1.6.1.6

Multiply $- 6$ by $- 6$ .

$4 x^{2} + 36 x + 81 - (64 y^{2} - 48 y - 48 y + 36) = 4$

Step 1.1.6.2

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

$4 x^{2} + 36 x + 81 - (64 y^{2} - 96 y + 36) = 4$

Step 1.1.7

Apply the distributive property.

$4 x^{2} + 36 x + 81 - (64 y^{2}) - (- 96 y) - 1 \cdot 36 = 4$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Multiply $64$ by $- 1$ .

$4 x^{2} + 36 x + 81 - 64 y^{2} - (- 96 y) - 1 \cdot 36 = 4$

Step 1.1.8.2

Multiply $- 96$ by $- 1$ .

$4 x^{2} + 36 x + 81 - 64 y^{2} + 96 y - 1 \cdot 36 = 4$

Step 1.1.8.3

Multiply $- 1$ by $36$ .

$4 x^{2} + 36 x + 81 - 64 y^{2} + 96 y - 36 = 4$

Step 1.2

Simplify the expression.

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Step 1.2.1

Subtract $36$ from $81$ .

$4 x^{2} + 36 x - 64 y^{2} + 96 y + 45 = 4$

Step 1.2.2

Move $36 x$ .

$4 x^{2} - 64 y^{2} + 36 x + 96 y + 45 = 4$

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract $4$ from both sides of the equation.

$4 x^{2} - 64 y^{2} + 36 x + 96 y + 45 - 4 = 0$

Step 2.2

Subtract $4$ from $45$ .

$4 x^{2} - 64 y^{2} + 36 x + 96 y + 41 = 0$

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