Algebra Examples

$2 x^{2} D - 5 y + 1 = 4 x + 2 y^{2} D$

Step 1

Subtract $2 y^{2} D$ from both sides of the equation.

$2 x^{2} D - 5 y + 1 - 2 y^{2} D = 4 x$

Step 2

Move all terms not containing $D$ to the right side of the equation.

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Step 2.1

Add $5 y$ to both sides of the equation.

$2 x^{2} D + 1 - 2 y^{2} D = 4 x + 5 y$

Step 2.2

Subtract $1$ from both sides of the equation.

$2 x^{2} D - 2 y^{2} D = 4 x + 5 y - 1$

Step 3

Factor $2 D$ out of $2 x^{2} D - 2 y^{2} D$ .

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Step 3.1

Factor $2 D$ out of $2 x^{2} D$ .

$2 D (x^{2}) - 2 y^{2} D = 4 x + 5 y - 1$

Step 3.2

Factor $2 D$ out of $- 2 y^{2} D$ .

$2 D (x^{2}) + 2 D (- y^{2}) = 4 x + 5 y - 1$

Step 3.3

Factor $2 D$ out of $2 D (x^{2}) + 2 D (- y^{2})$ .

$2 D (x^{2} - y^{2}) = 4 x + 5 y - 1$

Step 4

Factor.

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Step 4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = y$ .

$2 D ((x + y) (x - y)) = 4 x + 5 y - 1$

Step 4.2

Remove unnecessary parentheses.

$2 D (x + y) (x - y) = 4 x + 5 y - 1$

Step 5

Divide each term in $2 D (x + y) (x - y) = 4 x + 5 y - 1$ by $2 (x + y) (x - y)$ and simplify.

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Step 5.1

Divide each term in $2 D (x + y) (x - y) = 4 x + 5 y - 1$ by $2 (x + y) (x - y)$ .

$\frac{2 D (x + y) (x - y)}{2 (x + y) (x - y)} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 D (x + y) (x - y)}{2 (x + y) (x - y)} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2.1.2

Rewrite the expression.

$\frac{(D (x + y)) (x - y)}{(x + y) (x - y)} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2.2

Cancel the common factor of $x + y$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{D (x + y) (x - y)}{(x + y) (x - y)} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2.2.2

Rewrite the expression.

$\frac{(D) (x - y)}{x - y} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2.3

Cancel the common factor of $x - y$ .

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Step 5.2.3.1

Cancel the common factor.

$\frac{D (x - y)}{x - y} = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.2.3.2

Divide $D$ by $1$ .

$D = \frac{4 x}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Cancel the common factor of $4$ and $2$ .

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Step 5.3.1.1.1

Factor $2$ out of $4 x$ .

$D = \frac{2 (2 x)}{2 (x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.3.1.1.2

Cancel the common factors.

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Step 5.3.1.1.2.1

Factor $2$ out of $2 (x + y) (x - y)$ .

$D = \frac{2 (2 x)}{2 ((x + y) (x - y))} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.3.1.1.2.2

Cancel the common factor.

$D = \frac{2 (2 x)}{2 ((x + y) (x - y))} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.3.1.1.2.3

Rewrite the expression.

$D = \frac{2 x}{(x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} + \frac{- 1}{2 (x + y) (x - y)}$

Step 5.3.1.2

Move the negative in front of the fraction.

$D = \frac{2 x}{(x + y) (x - y)} + \frac{5 y}{2 (x + y) (x - y)} - \frac{1}{2 (x + y) (x - y)}$