Calculus Examples

$x^{2} + x y + y^{2} = 7$

Step 1

Subtract $7$ from both sides of the equation.

$x^{2} + x y + y^{2} - 7 = 0$

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 $a = 1$ , $b = y$ , and $c = y^{2} - 7$ into the quadratic formula and solve for $x$ .

$\frac{- y \pm \sqrt{y^{2} - 4 \cdot (1 \cdot (y^{2} - 7))}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Multiply $- 4$ by $1$ .

$x = \frac{- y \pm \sqrt{y^{2} - 4 \cdot (y^{2} - 7)}}{2 \cdot 1}$

Step 4.1.2

Apply the distributive property.

$x = \frac{- y \pm \sqrt{y^{2} - 4 y^{2} - 4 \cdot - 7}}{2 \cdot 1}$

Step 4.1.3

Multiply $- 4$ by $- 7$ .

$x = \frac{- y \pm \sqrt{y^{2} - 4 y^{2} + 28}}{2 \cdot 1}$

Step 4.1.4

Subtract $4 y^{2}$ from $y^{2}$ .

$x = \frac{- y \pm \sqrt{- 3 y^{2} + 28}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{- y \pm \sqrt{- 3 y^{2} + 28}}{2}$

Step 5

The final answer is the combination of both solutions.

$x = - \frac{y - \sqrt{- 3 y^{2} + 28}}{2}$

$x = - \frac{y + \sqrt{- 3 y^{2} + 28}}{2}$