Examples

${(2 x - 5)}^{2} - {(5 y - 4)}^{2} = 4$

Step 1

Simplify the left side ${(2 x - 5)}^{2} - {(5 y - 4)}^{2}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

$(2 x - 5) (2 x - 5) - {(5 y - 4)}^{2} = 4$

Step 1.1.2

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$2 x (2 x - 5) - 5 (2 x - 5) - {(5 y - 4)}^{2} = 4$

Step 1.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5) - {(5 y - 4)}^{2} = 4$

Step 1.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

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.

$2 \cdot (2 x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

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 $x$ .

$2 \cdot (2 (x \cdot x)) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot (2 x^{2}) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.1.4

Multiply $- 5$ by $2$ .

$4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.1.5

Multiply $2$ by $- 5$ .

$4 x^{2} - 10 x - 10 x - 5 \cdot - 5 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.1.6

Multiply $- 5$ by $- 5$ .

$4 x^{2} - 10 x - 10 x + 25 - {(5 y - 4)}^{2} = 4$

Step 1.1.3.2

Subtract $10 x$ from $- 10 x$ .

$4 x^{2} - 20 x + 25 - {(5 y - 4)}^{2} = 4$

Step 1.1.4

Rewrite ${(5 y - 4)}^{2}$ as $(5 y - 4) (5 y - 4)$ .

$4 x^{2} - 20 x + 25 - ((5 y - 4) (5 y - 4)) = 4$

Step 1.1.5

Expand $(5 y - 4) (5 y - 4)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (5 y (5 y - 4) - 4 (5 y - 4)) = 4$

Step 1.1.5.2

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (5 y (5 y) + 5 y \cdot - 4 - 4 (5 y - 4)) = 4$

Step 1.1.5.3

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (5 y (5 y) + 5 y \cdot - 4 - 4 (5 y) - 4 \cdot - 4) = 4$

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.

$4 x^{2} - 20 x + 25 - (5 \cdot (5 y \cdot y) + 5 y \cdot - 4 - 4 (5 y) - 4 \cdot - 4) = 4$

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 $y$ .

$4 x^{2} - 20 x + 25 - (5 \cdot (5 (y \cdot y)) + 5 y \cdot - 4 - 4 (5 y) - 4 \cdot - 4) = 4$

Step 1.1.6.1.2.2

Multiply $y$ by $y$ .

$4 x^{2} - 20 x + 25 - (5 \cdot (5 y^{2}) + 5 y \cdot - 4 - 4 (5 y) - 4 \cdot - 4) = 4$

Step 1.1.6.1.3

Multiply $5$ by $5$ .

$4 x^{2} - 20 x + 25 - (25 y^{2} + 5 y \cdot - 4 - 4 (5 y) - 4 \cdot - 4) = 4$

Step 1.1.6.1.4

Multiply $- 4$ by $5$ .

$4 x^{2} - 20 x + 25 - (25 y^{2} - 20 y - 4 (5 y) - 4 \cdot - 4) = 4$

Step 1.1.6.1.5

Multiply $5$ by $- 4$ .

$4 x^{2} - 20 x + 25 - (25 y^{2} - 20 y - 20 y - 4 \cdot - 4) = 4$

Step 1.1.6.1.6

Multiply $- 4$ by $- 4$ .

$4 x^{2} - 20 x + 25 - (25 y^{2} - 20 y - 20 y + 16) = 4$

Step 1.1.6.2

Subtract $20 y$ from $- 20 y$ .

$4 x^{2} - 20 x + 25 - (25 y^{2} - 40 y + 16) = 4$

Step 1.1.7

Apply the distributive property.

$4 x^{2} - 20 x + 25 - (25 y^{2}) - (- 40 y) - 1 \cdot 16 = 4$

Step 1.1.8

Simplify.

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Step 1.1.8.1

Multiply $25$ by $- 1$ .

$4 x^{2} - 20 x + 25 - 25 y^{2} - (- 40 y) - 1 \cdot 16 = 4$

Step 1.1.8.2

Multiply $- 40$ by $- 1$ .

$4 x^{2} - 20 x + 25 - 25 y^{2} + 40 y - 1 \cdot 16 = 4$

Step 1.1.8.3

Multiply $- 1$ by $16$ .

$4 x^{2} - 20 x + 25 - 25 y^{2} + 40 y - 16 = 4$

Step 1.2

Simplify the expression.

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Step 1.2.1

Subtract $16$ from $25$ .

$4 x^{2} - 20 x - 25 y^{2} + 40 y + 9 = 4$

Step 1.2.2

Move $- 20 x$ .

$4 x^{2} - 25 y^{2} - 20 x + 40 y + 9 = 4$

Step 2

Set the equation equal to zero.

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Step 2.1

Subtract $4$ from both sides of the equation.

$4 x^{2} - 25 y^{2} - 20 x + 40 y + 9 - 4 = 0$

Step 2.2

Subtract $4$ from $9$ .

$4 x^{2} - 25 y^{2} - 20 x + 40 y + 5 = 0$

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