Algebra Examples

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

Step 1

Solve for $y$ .

Tap for more steps...

Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

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

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

Step 1.3

Simplify.

Tap for more steps...

Step 1.3.1

Simplify the numerator.

Tap for more steps...

Step 1.3.1.1

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

Tap for more steps...

Step 1.3.1.1.1

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

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

Step 1.3.1.1.2

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

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

Step 1.3.1.1.3

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

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

Step 1.3.1.2

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

Tap for more steps...

Step 1.3.1.2.1

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

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

Step 1.3.1.2.2

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

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

Step 1.3.1.2.3

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

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

Step 1.3.1.3

Add $1$ and $2$ .

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

Step 1.3.1.4

Multiply $- 4$ by $- 2$ .

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

Step 1.3.1.5

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

Tap for more steps...

Step 1.3.1.5.1

Factor $4$ out of $8$ .

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

Step 1.3.1.5.2

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

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

Step 1.3.1.5.3

Add parentheses.

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

Step 1.3.1.6

Pull terms out from under the radical.

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

Step 1.3.2

Multiply $2$ by $- 2$ .

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

Step 1.3.3

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

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

Step 1.3.4

Move the negative in front of the fraction.

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

Simplify the numerator.

Tap for more steps...

Step 1.4.1.1

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

Tap for more steps...

Step 1.4.1.1.1

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

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

Step 1.4.1.1.2

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

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

Step 1.4.1.1.3

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

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

Step 1.4.1.2

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

Tap for more steps...

Step 1.4.1.2.1

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

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

Step 1.4.1.2.2

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

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

Step 1.4.1.2.3

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

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

Step 1.4.1.3

Add $1$ and $2$ .

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

Step 1.4.1.4

Multiply $- 4$ by $- 2$ .

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

Step 1.4.1.5

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

Tap for more steps...

Step 1.4.1.5.1

Factor $4$ out of $8$ .

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

Step 1.4.1.5.2

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

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

Step 1.4.1.5.3

Add parentheses.

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

Step 1.4.1.6

Pull terms out from under the radical.

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

Step 1.4.2

Multiply $2$ by $- 2$ .

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

Step 1.4.3

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

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

Step 1.4.4

Move the negative in front of the fraction.

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

Step 1.4.5

Change the $\pm$ to $+$ .

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

Step 1.5

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

Tap for more steps...

Step 1.5.1

Simplify the numerator.

Tap for more steps...

Step 1.5.1.1

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

Tap for more steps...

Step 1.5.1.1.1

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

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

Step 1.5.1.1.2

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

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

Step 1.5.1.1.3

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

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

Step 1.5.1.2

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

Tap for more steps...

Step 1.5.1.2.1

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

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

Step 1.5.1.2.2

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

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

Step 1.5.1.2.3

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

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

Step 1.5.1.3

Add $1$ and $2$ .

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

Step 1.5.1.4

Multiply $- 4$ by $- 2$ .

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

Step 1.5.1.5

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

Tap for more steps...

Step 1.5.1.5.1

Factor $4$ out of $8$ .

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

Step 1.5.1.5.2

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

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

Step 1.5.1.5.3

Add parentheses.

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

Step 1.5.1.6

Pull terms out from under the radical.

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

Step 1.5.2

Multiply $2$ by $- 2$ .

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

Step 1.5.3

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

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

Step 1.5.4

Move the negative in front of the fraction.

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

Step 1.5.5

Change the $\pm$ to $-$ .

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

Step 1.6

The final answer is the combination of both solutions.

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

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

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

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

Step 2

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

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

Step 3

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

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

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

Step 4

Divide $2$ by $2$ .

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

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

Step 5

Apply the distributive property.

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

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

Step 6

Multiply $- 1$ by $1$ .

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

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

Step 7

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

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

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

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Divide $2$ by $2$ .

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

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

Step 8.2

Move the negative in front of the fraction.

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

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

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

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

Step 9

Simplify by multiplying through.

Tap for more steps...

Step 9.1

Apply the distributive property.

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

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

Step 9.2

Multiply $- 1$ by $1$ .

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

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

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

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

Step 10

Reorder terms.

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

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

Step 11

Remove parentheses.

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

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

Step 12