Write as a Function of y 2x^2+2x+y=4y

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

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $4 y$ from both sides of the equation.

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

Subtract $4 y$ from $y$ .

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

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

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

Step 4

Simplify.

Simplify the numerator.

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2^{2}$ .

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

Factor $2$ out of $- 4 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot - 3 y)$ .

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

Factor $2$ out of $2 - 2 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot - 3 y)$ .

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

Multiply $- 1 \cdot 2 \cdot - 3$ .

Multiply $- 1$ by $2$ .

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

Multiply $- 2$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 + 6 y))}}{2 \cdot 2}$

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (1 + 6 y))}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (1 + 6 y)}}{2 \cdot 2}$

Rewrite $4 (1 + 6 y)$ as $2^{2} (1^{2} + 6 y)$ .

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1 + 6 y)}}{2 \cdot 2}$

Rewrite $1$ as $1^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1^{2} + 6 y)}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{1^{2} + 6 y}}{2 \cdot 2}$

One to any power is one.

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$

Simplify $\frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$ .

$x = \frac{- 1 \pm \sqrt{1 + 6 y}}{2}$

Step 5

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

Simplify the numerator.

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2^{2}$ .

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

Factor $2$ out of $- 4 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot - 3 y)$ .

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

Factor $2$ out of $2 - 2 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot - 3 y)$ .

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

Multiply $- 1 \cdot 2 \cdot - 3$ .

Multiply $- 1$ by $2$ .

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

Multiply $- 2$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 + 6 y))}}{2 \cdot 2}$

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (1 + 6 y))}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (1 + 6 y)}}{2 \cdot 2}$

Rewrite $4 (1 + 6 y)$ as $2^{2} (1^{2} + 6 y)$ .

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1 + 6 y)}}{2 \cdot 2}$

Rewrite $1$ as $1^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1^{2} + 6 y)}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{1^{2} + 6 y}}{2 \cdot 2}$

One to any power is one.

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$

Simplify $\frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$ .

$x = \frac{- 1 \pm \sqrt{1 + 6 y}}{2}$

Change the $\pm$ to $+$ .

$x = \frac{- 1 + \sqrt{1 + 6 y}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 + \sqrt{1 + 6 y}}{2}$

Factor $- 1$ out of $\sqrt{1 + 6 y}$ .

$x = \frac{- 1 \cdot 1 - 1 (- \sqrt{1 + 6 y})}{2}$

Factor $- 1$ out of $- 1 (1) - 1 (- \sqrt{1 + 6 y})$ .

$x = \frac{- 1 (1 - \sqrt{1 + 6 y})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 - \sqrt{1 + 6 y}}{2}$

Step 6

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

Simplify the numerator.

Factor $2$ out of $2^{2} - 4 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2^{2}$ .

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

Factor $2$ out of $- 4 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 \cdot 2 + 2 (- 2 \cdot 2 \cdot - 3 y)$ .

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

Factor $2$ out of $2 - 2 \cdot 2 \cdot - 3 y$ .

Factor $2$ out of $2$ .

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

Factor $2$ out of $- 2 \cdot 2 \cdot - 3 y$ .

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

Factor $2$ out of $2 (1) + 2 (- 1 \cdot 2 \cdot - 3 y)$ .

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

Multiply $- 1 \cdot 2 \cdot - 3$ .

Multiply $- 1$ by $2$ .

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

Multiply $- 2$ by $- 3$ .

$x = \frac{- 2 \pm \sqrt{2 (2 (1 + 6 y))}}{2 \cdot 2}$

$x = \frac{- 2 \pm \sqrt{2 \cdot (2 (1 + 6 y))}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 (1 + 6 y)}}{2 \cdot 2}$

Rewrite $4 (1 + 6 y)$ as $2^{2} (1^{2} + 6 y)$ .

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1 + 6 y)}}{2 \cdot 2}$

Rewrite $1$ as $1^{2}$ .

$x = \frac{- 2 \pm \sqrt{2^{2} (1^{2} + 6 y)}}{2 \cdot 2}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm 2 \sqrt{1^{2} + 6 y}}{2 \cdot 2}$

One to any power is one.

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$

Simplify $\frac{- 2 \pm 2 \sqrt{1 + 6 y}}{4}$ .

$x = \frac{- 1 \pm \sqrt{1 + 6 y}}{2}$

Change the $\pm$ to $-$ .

$x = \frac{- 1 - \sqrt{1 + 6 y}}{2}$

Factor $- 1$ out of $- 1 - \sqrt{1 + 6 y}$ .

Reorder $1$ and $6 y$ .

$x = \frac{- 1 - \sqrt{6 y + 1}}{2}$

Rewrite $- 1$ as $- 1 (1)$ .

$x = \frac{- 1 \cdot 1 - \sqrt{6 y + 1}}{2}$

Factor $- 1$ out of $- \sqrt{6 y + 1}$ .

$x = \frac{- 1 \cdot 1 - (\sqrt{6 y + 1})}{2}$

Factor $- 1$ out of $- 1 (1) - (\sqrt{6 y + 1})$ .

$x = \frac{- 1 (1 + \sqrt{6 y + 1})}{2}$

Rewrite $- 1 (1 + \sqrt{6 y + 1})$ as $- (1 + \sqrt{6 y + 1})$ .

$x = \frac{- (1 + \sqrt{6 y + 1})}{2}$

Move the negative in front of the fraction.

$x = - \frac{1 + \sqrt{6 y + 1}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{1 + 6 y}}{2}$

$x = - \frac{1 + \sqrt{6 y + 1}}{2}$

Write as a Function of y 2x^2+2x+y=4y

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