Calculus Examples

$X + y = X y (X - y)$

Step 1

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$X y (X - y) = X + y$

Step 2

Simplify $X y (X - y)$ .

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Step 2.1

Rewrite.

$0 + 0 + X y (X - y) = X + y$

Step 2.2

Simplify by adding zeros.

$X y (X - y) = X + y$

Step 2.3

Apply the distributive property.

$X y X + X y (- y) = X + y$

Step 2.4

Multiply $X$ by $X$ by adding the exponents.

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Step 2.4.1

Move $X$ .

$X \cdot X y + X y (- y) = X + y$

Step 2.4.2

Multiply $X$ by $X$ .

$X^{2} y + X y (- y) = X + y$

Step 2.5

Rewrite using the commutative property of multiplication.

$X^{2} y - X y \cdot y = X + y$

Step 2.6

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

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Step 2.6.1

Move $y$ .

$X^{2} y - X (y \cdot y) = X + y$

Step 2.6.2

Multiply $y$ by $y$ .

$X^{2} y - X y^{2} = X + y$

Step 3

Subtract $y$ from both sides of the equation.

$X^{2} y - X y^{2} - y = X$

Step 4

Subtract $X$ from both sides of the equation.

$X^{2} y - X y^{2} - y - X = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - X$ , $b = X^{2} - 1$ , and $c = - X$ into the quadratic formula and solve for $y$ .

$\frac{- (X^{2} - 1) \pm \sqrt{{(X^{2} - 1)}^{2} - 4 \cdot (- X \cdot (- X))}}{2 (- X)}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$y = \frac{- X^{2} + 1 \pm \sqrt{{(X^{2} - 1)}^{2} - 4 \cdot (- 1 X) \cdot (- X)}}{2 (- X)}$

Step 7.1.2

Multiply $- 1$ by $- 1$ .

$y = \frac{- X^{2} + 1 \pm \sqrt{{(X^{2} - 1)}^{2} - 4 \cdot (- 1 X) \cdot (- X)}}{2 (- X)}$

Step 7.1.3

Rewrite $- 4 X \cdot (- X)$ as ${(2 X)}^{2}$ .

$y = \frac{- X^{2} + 1 \pm \sqrt{{(X^{2} - 1)}^{2} - {(2 X)}^{2}}}{2 (- X)}$

Step 7.1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = X^{2} - 1$ and $b = 2 X$ .

$y = \frac{- X^{2} + 1 \pm \sqrt{(X^{2} - 1 + 2 X) (X^{2} - 1 - (2 X))}}{2 (- X)}$

Step 7.1.5

Simplify.

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Step 7.1.5.1

Reorder terms.

$y = \frac{- X^{2} + 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 1 - (2 X))}}{2 (- X)}$

Step 7.1.5.2

Multiply $2$ by $- 1$ .

$y = \frac{- X^{2} + 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 1 - 2 X)}}{2 (- X)}$

Step 7.1.5.3

Reorder terms.

$y = \frac{- X^{2} + 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{2 (- X)}$

Step 7.2

Multiply $- 1$ by $2$ .

$y = \frac{- X^{2} + 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{- 2 X}$

Step 7.3

Simplify $\frac{- X^{2} + 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{- 2 X}$ .

$y = \frac{X^{2} - 1 \pm \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{2 X}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{X^{2} - 1 + \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{2 X}$

$y = \frac{X^{2} - 1 - \sqrt{(X^{2} + 2 X - 1) (X^{2} - 2 X - 1)}}{2 X}$