Algebra Examples

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

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

Step 1

Simplify the left side

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

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite

{(2 x - 5)}^{2}

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

(2 x - 5) (2 x - 5)

$(2 x - 5) (2 x - 5)$ .

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

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

Step 1.1.2

Expand

(2 x - 5) (2 x - 5)

$(2 x - 5) (2 x - 5)$ 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

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

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

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

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

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

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

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

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

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

Step 1.1.3.1.2

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 1.1.3.1.2.1

Move

x

$x$ .

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

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

Step 1.1.3.1.2.2

Multiply

x

$x$ by

x

$x$ .

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

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

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

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

Step 1.1.3.1.3

Multiply

2

$2$ by

2

$2$ .

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

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

Step 1.1.3.1.4

Multiply

- 5

$- 5$ by

2

$2$ .

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

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

Step 1.1.3.1.5

Multiply

2

$2$ by

- 5

$- 5$ .

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

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

Step 1.1.3.1.6

Multiply

- 5

$- 5$ by

- 5

$- 5$ .

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

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

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

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

Step 1.1.3.2

Subtract

10 x

$10 x$ from

- 10 x

$- 10 x$ .

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

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

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

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

Step 1.1.4

Rewrite

{(5 y - 4)}^{2}

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

(5 y - 4) (5 y - 4)

$(5 y - 4) (5 y - 4)$ .

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

$(5 y - 4) (5 y - 4)$ 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

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

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

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

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

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

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

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

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

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

Step 1.1.6.1.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

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Step 1.1.6.1.2.1

Move

y

$y$ .

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

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

Step 1.1.6.1.2.2

Multiply

y

$y$ by

y

$y$ .

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

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

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

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

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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