Write as a Function of y x^2-xy=4

$x^{2} - x y = 4$

Step 1

Subtract $4$ from both sides of the equation.

$x^{2} - x y - 4 = 0$

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 $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

Apply the product rule to $- y$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $y^{2}$ by $1$ .

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

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Apply the product rule to $- y$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $y^{2}$ by $1$ .

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

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{y + \sqrt{y^{2} + 16}}{2}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Apply the product rule to $- y$ .

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

Raise $- 1$ to the power of $2$ .

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

Multiply $y^{2}$ by $1$ .

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

Multiply $- 4 \cdot 1 \cdot - 4$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $- 4$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{y \pm \sqrt{y^{2} + 16}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{y - \sqrt{y^{2} + 16}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{y + \sqrt{y^{2} + 16}}{2}$

$x = \frac{y - \sqrt{y^{2} + 16}}{2}$

Write as a Function of y x^2-xy=4

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