Calculus Examples

$x^{2} - 3 x y + y^{2} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$x^{2} - 3 x y + y^{2} - 1 = 0$

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 $a = 1$ , $b = - 3 x$ , and $c = x^{2} - 1$ into the quadratic formula and solve for $y$ .

$\frac{3 x \pm \sqrt{{(- 3 x)}^{2} - 4 \cdot (1 \cdot (x^{2} - 1))}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the product rule to $- 3 x$ .

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

Step 4.1.2

Raise $- 3$ to the power of $2$ .

$y = \frac{3 x \pm \sqrt{9 x^{2} - 4 \cdot 1 \cdot (x^{2} - 1)}}{2 \cdot 1}$

Step 4.1.3

Multiply $- 4$ by $1$ .

$y = \frac{3 x \pm \sqrt{9 x^{2} - 4 \cdot (x^{2} - 1)}}{2 \cdot 1}$

Step 4.1.4

Apply the distributive property.

$y = \frac{3 x \pm \sqrt{9 x^{2} - 4 x^{2} - 4 \cdot - 1}}{2 \cdot 1}$

Step 4.1.5

Multiply $- 4$ by $- 1$ .

$y = \frac{3 x \pm \sqrt{9 x^{2} - 4 x^{2} + 4}}{2 \cdot 1}$

Step 4.1.6

Subtract $4 x^{2}$ from $9 x^{2}$ .

$y = \frac{3 x \pm \sqrt{5 x^{2} + 4}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$y = \frac{3 x \pm \sqrt{5 x^{2} + 4}}{2}$

Step 5

The final answer is the combination of both solutions.

$y = \frac{3 x + \sqrt{5 x^{2} + 4}}{2}$

$y = \frac{3 x - \sqrt{5 x^{2} + 4}}{2}$