Examples

$2 {(x - 1)}^{2} - {(3 y - 4)}^{2} = 25$

Step 1

Simplify the left side $2 {(x - 1)}^{2} - {(3 y - 4)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$2 ((x - 1) (x - 1)) - {(3 y - 4)}^{2} = 25$

Step 1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$2 (x (x - 1) - 1 (x - 1)) - {(3 y - 4)}^{2} = 25$

Step 1.1.2.2

Apply the distributive property.

$2 (x \cdot x + x \cdot - 1 - 1 (x - 1)) - {(3 y - 4)}^{2} = 25$

Step 1.1.2.3

Apply the distributive property.

$2 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) - {(3 y - 4)}^{2} = 25$

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

Multiply $x$ by $x$ .

$2 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.3.1.2

Move $- 1$ to the left of $x$ .

$2 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$2 (x^{2} - x - 1 x - 1 \cdot - 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$2 (x^{2} - x - x - 1 \cdot - 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$2 (x^{2} - x - x + 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.3.2

Subtract $x$ from $- x$ .

$2 (x^{2} - 2 x + 1) - {(3 y - 4)}^{2} = 25$

Step 1.1.4

Apply the distributive property.

$2 x^{2} + 2 (- 2 x) + 2 \cdot 1 - {(3 y - 4)}^{2} = 25$

Step 1.1.5

Simplify.

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Step 1.1.5.1

Multiply $- 2$ by $2$ .

$2 x^{2} - 4 x + 2 \cdot 1 - {(3 y - 4)}^{2} = 25$

Step 1.1.5.2

Multiply $2$ by $1$ .

$2 x^{2} - 4 x + 2 - {(3 y - 4)}^{2} = 25$

Step 1.1.6

Rewrite ${(3 y - 4)}^{2}$ as $(3 y - 4) (3 y - 4)$ .

$2 x^{2} - 4 x + 2 - ((3 y - 4) (3 y - 4)) = 25$

Step 1.1.7

Expand $(3 y - 4) (3 y - 4)$ using the FOIL Method.

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Step 1.1.7.1

Apply the distributive property.

$2 x^{2} - 4 x + 2 - (3 y (3 y - 4) - 4 (3 y - 4)) = 25$

Step 1.1.7.2

Apply the distributive property.

$2 x^{2} - 4 x + 2 - (3 y (3 y) + 3 y \cdot - 4 - 4 (3 y - 4)) = 25$

Step 1.1.7.3

Apply the distributive property.

$2 x^{2} - 4 x + 2 - (3 y (3 y) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8

Simplify and combine like terms.

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Rewrite using the commutative property of multiplication.

$2 x^{2} - 4 x + 2 - (3 \cdot (3 y \cdot y) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8.1.2

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

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Step 1.1.8.1.2.1

Move $y$ .

$2 x^{2} - 4 x + 2 - (3 \cdot (3 (y \cdot y)) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8.1.2.2

Multiply $y$ by $y$ .

$2 x^{2} - 4 x + 2 - (3 \cdot (3 y^{2}) + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8.1.3

Multiply $3$ by $3$ .

$2 x^{2} - 4 x + 2 - (9 y^{2} + 3 y \cdot - 4 - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8.1.4

Multiply $- 4$ by $3$ .

$2 x^{2} - 4 x + 2 - (9 y^{2} - 12 y - 4 (3 y) - 4 \cdot - 4) = 25$

Step 1.1.8.1.5

Multiply $3$ by $- 4$ .

$2 x^{2} - 4 x + 2 - (9 y^{2} - 12 y - 12 y - 4 \cdot - 4) = 25$

Step 1.1.8.1.6

Multiply $- 4$ by $- 4$ .

$2 x^{2} - 4 x + 2 - (9 y^{2} - 12 y - 12 y + 16) = 25$

Step 1.1.8.2

Subtract $12 y$ from $- 12 y$ .

$2 x^{2} - 4 x + 2 - (9 y^{2} - 24 y + 16) = 25$

Step 1.1.9

Apply the distributive property.

$2 x^{2} - 4 x + 2 - (9 y^{2}) - (- 24 y) - 1 \cdot 16 = 25$

Step 1.1.10

Simplify.

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Step 1.1.10.1

Multiply $9$ by $- 1$ .

$2 x^{2} - 4 x + 2 - 9 y^{2} - (- 24 y) - 1 \cdot 16 = 25$

Step 1.1.10.2

Multiply $- 24$ by $- 1$ .

$2 x^{2} - 4 x + 2 - 9 y^{2} + 24 y - 1 \cdot 16 = 25$

Step 1.1.10.3

Multiply $- 1$ by $16$ .

$2 x^{2} - 4 x + 2 - 9 y^{2} + 24 y - 16 = 25$

Step 1.2

Simplify the expression.

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Step 1.2.1

Subtract $16$ from $2$ .

$2 x^{2} - 4 x - 9 y^{2} + 24 y - 14 = 25$

Step 1.2.2

Move $- 4 x$ .

$2 x^{2} - 9 y^{2} - 4 x + 24 y - 14 = 25$

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract $25$ from both sides of the equation.

$2 x^{2} - 9 y^{2} - 4 x + 24 y - 14 - 25 = 0$

Step 2.2

Subtract $25$ from $- 14$ .

$2 x^{2} - 9 y^{2} - 4 x + 24 y - 39 = 0$

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