Examples

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

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

Step 1

Simplify the left side

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

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

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Rewrite

{(x - 1)}^{2}

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

(x - 1) (x - 1)

$(x - 1) (x - 1)$ .

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

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

Step 1.1.2

Expand

(x - 1) (x - 1)

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

Tap for more steps...

Step 1.1.2.1

Apply the distributive property.

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

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

Step 1.1.2.2

Apply the distributive property.

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

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

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

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

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

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

Step 1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

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

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

Step 1.1.3.1.2

Move

- 1

$- 1$ to the left of

x

$x$ .

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

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

Step 1.1.3.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

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

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

Step 1.1.3.1.4

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

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

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

Step 1.1.3.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

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

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

Step 1.1.3.2

Subtract

x

$x$ from

- x

$- x$ .

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

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

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

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

Step 1.1.4

Apply the distributive property.

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

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

Step 1.1.5

Simplify.

Tap for more steps...

Step 1.1.5.1

Multiply

- 2

$- 2$ by

2

$2$ .

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

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

Step 1.1.5.2

Multiply

2

$2$ by

1

$1$ .

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

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

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

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

Step 1.1.6

Rewrite

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

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

(3 y - 4) (3 y - 4)

$(3 y - 4) (3 y - 4)$ .

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

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

Step 1.1.7

Expand

(3 y - 4) (3 y - 4)

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

Tap for more steps...

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

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

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

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

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

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

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

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

Step 1.1.8

Simplify and combine like terms.

Tap for more steps...

Step 1.1.8.1

Simplify each term.

Tap for more steps...

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

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

Step 1.1.8.1.2

Multiply

y

$y$ by

y

$y$ by adding the exponents.

Tap for more steps...

Step 1.1.8.1.2.1

Move

y

$y$ .

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

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

Step 1.1.8.1.2.2

Multiply

y

$y$ by

y

$y$ .

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

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

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

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

Step 1.1.8.1.3

Multiply

3

$3$ by

3

$3$ .

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

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

Step 1.1.8.1.4

Multiply

- 4

$- 4$ by

3

$3$ .

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

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

Step 1.1.8.1.5

Multiply

3

$3$ by

- 4

$- 4$ .

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

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

Step 1.1.8.1.6

Multiply

- 4

$- 4$ by

- 4

$- 4$ .

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

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

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

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

Step 1.1.8.2

Subtract

12 y

$12 y$ from

- 12 y

$- 12 y$ .

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

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

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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

$- 14$ .

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

Enter YOUR Problem