Calculus Examples

$x^{2} + x y - y^{2} = - 1$

Step 1

Add $1$ to both sides of the equation.

$x^{2} + x y - y^{2} + 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 4$ by $- 1$ .

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Multiply $4$ by $1$ .

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

Step 4.1.4

Add $x^{2}$ and $4 x^{2}$ .

$y = \frac{- x \pm \sqrt{5 x^{2} + 4}}{2 \cdot - 1}$

Step 4.2

Multiply $2$ by $- 1$ .

$y = \frac{- x \pm \sqrt{5 x^{2} + 4}}{- 2}$

Step 4.3

Simplify $\frac{- x \pm \sqrt{5 x^{2} + 4}}{- 2}$ .

$y = \frac{x \pm \sqrt{5 x^{2} + 4}}{2}$

Step 5

The final answer is the combination of both solutions.

$y = \frac{x + \sqrt{5 x^{2} + 4}}{2}$

$y = \frac{x - \sqrt{5 x^{2} + 4}}{2}$