Write in Standard Form -2y^2+x-4y+1=0

$- 2 y^{2} + x - 4 y + 1 = 0$

Solve for $y$ .

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 2$ , $b = - 4$ , and $c = x + 1$ into the quadratic formula and solve for $y$ .

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

Simplify.

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 2 \cdot (x + 1)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Factor $- 4$ out of $- 4 \cdot - 2 \cdot (x + 1)$ .

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

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 2 \cdot (x + 1))$ .

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

Factor $- 2$ out of $- 4 - 2 \cdot (x + 1)$ .

Reorder $- 4$ and $- 2 \cdot (x + 1)$ .

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

Factor $- 2$ out of $- 4$ .

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

Factor $- 2$ out of $- 2 (x + 1) - 2 (2)$ .

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

Add $1$ and $2$ .

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

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x + 3)}}{2 \cdot - 2}$

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 2$ .

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

Simplify $\frac{4 \pm 2 \sqrt{2 (x + 3)}}{- 4}$ .

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

Move the negative in front of the fraction.

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

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 2 \cdot (x + 1)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Factor $- 4$ out of $- 4 \cdot - 2 \cdot (x + 1)$ .

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

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 2 \cdot (x + 1))$ .

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

Factor $- 2$ out of $- 4 - 2 \cdot (x + 1)$ .

Reorder $- 4$ and $- 2 \cdot (x + 1)$ .

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

Factor $- 2$ out of $- 4$ .

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

Factor $- 2$ out of $- 2 (x + 1) - 2 (2)$ .

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

Add $1$ and $2$ .

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

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x + 3)}}{2 \cdot - 2}$

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 2$ .

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

Simplify $\frac{4 \pm 2 \sqrt{2 (x + 3)}}{- 4}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 2 \cdot (x + 1)$ .

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Factor $- 4$ out of $- 4 \cdot - 2 \cdot (x + 1)$ .

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

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 2 \cdot (x + 1))$ .

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

Factor $- 2$ out of $- 4 - 2 \cdot (x + 1)$ .

Reorder $- 4$ and $- 2 \cdot (x + 1)$ .

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

Factor $- 2$ out of $- 4$ .

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

Factor $- 2$ out of $- 2 (x + 1) - 2 (2)$ .

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

Add $1$ and $2$ .

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

Multiply $- 4$ by $- 2$ .

$y = \frac{4 \pm \sqrt{8 (x + 3)}}{2 \cdot - 2}$

Rewrite $8 (x + 3)$ as $2^{2} \cdot (2 (x + 3))$ .

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 2$ .

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

Simplify $\frac{4 \pm 2 \sqrt{2 (x + 3)}}{- 4}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

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

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

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

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$y = a x^{2} + b x + c$

Split the fraction $\frac{2 + \sqrt{2 (x + 3)}}{2}$ into two fractions.

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

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

Divide $2$ by $2$ .

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

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

Apply the distributive property.

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

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

Multiply $- 1$ by $1$ .