Write in Standard Form 2y^2-5x+12y+23=0

$2 y^{2} - 5 x + 12 y + 23 = 0$

Step 1

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 $a = 2$ , $b = 12$ , and $c = - 5 x + 23$ into the quadratic formula and solve for $y$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (2 \cdot (- 5 x + 23))}}{2 \cdot 2}$

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 2 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 5 x) - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 5$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 8$ by $23$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 184}}{2 \cdot 2}$

Subtract $184$ from $144$ .

$y = \frac{- 12 \pm \sqrt{40 x - 40}}{2 \cdot 2}$

Factor $40$ out of $40 x - 40$ .

Factor $40$ out of $40 x$ .

$y = \frac{- 12 \pm \sqrt{40 (x) - 40}}{2 \cdot 2}$

Factor $40$ out of $- 40$ .

$y = \frac{- 12 \pm \sqrt{40 (x) + 40 (- 1)}}{2 \cdot 2}$

Factor $40$ out of $40 (x) + 40 (- 1)$ .

$y = \frac{- 12 \pm \sqrt{40 (x - 1)}}{2 \cdot 2}$

Rewrite $40 (x - 1)$ as $2^{2} \cdot (10 (x - 1))$ .

Factor $4$ out of $40$ .

$y = \frac{- 12 \pm \sqrt{4 (10) (x - 1)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{10 (x - 1)}}{2}$

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 2 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 5 x) - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 5$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 8$ by $23$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 184}}{2 \cdot 2}$

Subtract $184$ from $144$ .

$y = \frac{- 12 \pm \sqrt{40 x - 40}}{2 \cdot 2}$

Factor $40$ out of $40 x - 40$ .

Factor $40$ out of $40 x$ .

$y = \frac{- 12 \pm \sqrt{40 (x) - 40}}{2 \cdot 2}$

Factor $40$ out of $- 40$ .

$y = \frac{- 12 \pm \sqrt{40 (x) + 40 (- 1)}}{2 \cdot 2}$

Factor $40$ out of $40 (x) + 40 (- 1)$ .

$y = \frac{- 12 \pm \sqrt{40 (x - 1)}}{2 \cdot 2}$

Rewrite $40 (x - 1)$ as $2^{2} \cdot (10 (x - 1))$ .

Factor $4$ out of $40$ .

$y = \frac{- 12 \pm \sqrt{4 (10) (x - 1)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{10 (x - 1)}}{2}$

Change the $\pm$ to $+$ .

$y = \frac{- 6 + \sqrt{10 (x - 1)}}{2}$

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 + \sqrt{10 (x - 1)}}{2}$

Factor $- 1$ out of $\sqrt{10 (x - 1)}$ .

$y = \frac{- 1 \cdot 6 - 1 (- \sqrt{10 (x - 1)})}{2}$

Factor $- 1$ out of $- 1 (6) - 1 (- \sqrt{10 (x - 1)})$ .

$y = \frac{- 1 (6 - \sqrt{10 (x - 1)})}{2}$

Move the negative in front of the fraction.

$y = - \frac{6 - \sqrt{10 (x - 1)}}{2}$

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

Simplify the numerator.

Raise $12$ to the power of $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 4 \cdot 2 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Multiply $- 4$ by $2$ .

$y = \frac{- 12 \pm \sqrt{144 - 8 \cdot (- 5 x + 23)}}{2 \cdot 2}$

Apply the distributive property.

$y = \frac{- 12 \pm \sqrt{144 - 8 (- 5 x) - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 5$ by $- 8$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 8 \cdot 23}}{2 \cdot 2}$

Multiply $- 8$ by $23$ .

$y = \frac{- 12 \pm \sqrt{144 + 40 x - 184}}{2 \cdot 2}$

Subtract $184$ from $144$ .

$y = \frac{- 12 \pm \sqrt{40 x - 40}}{2 \cdot 2}$

Factor $40$ out of $40 x - 40$ .

Factor $40$ out of $40 x$ .

$y = \frac{- 12 \pm \sqrt{40 (x) - 40}}{2 \cdot 2}$

Factor $40$ out of $- 40$ .

$y = \frac{- 12 \pm \sqrt{40 (x) + 40 (- 1)}}{2 \cdot 2}$

Factor $40$ out of $40 (x) + 40 (- 1)$ .

$y = \frac{- 12 \pm \sqrt{40 (x - 1)}}{2 \cdot 2}$

Rewrite $40 (x - 1)$ as $2^{2} \cdot (10 (x - 1))$ .

Factor $4$ out of $40$ .

$y = \frac{- 12 \pm \sqrt{4 (10) (x - 1)}}{2 \cdot 2}$

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

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Add parentheses.

$y = \frac{- 12 \pm \sqrt{2^{2} \cdot (10 (x - 1))}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$

Simplify $\frac{- 12 \pm 2 \sqrt{10 (x - 1)}}{4}$ .

$y = \frac{- 6 \pm \sqrt{10 (x - 1)}}{2}$

Change the $\pm$ to $-$ .

$y = \frac{- 6 - \sqrt{10 (x - 1)}}{2}$

Factor $- 1$ out of $- 6 - \sqrt{10 (x - 1)}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$y = \frac{- 1 \cdot 6 - \sqrt{10 (x - 1)}}{2}$

Factor $- 1$ out of $- \sqrt{10 (x - 1)}$ .

$y = \frac{- 1 \cdot 6 - (\sqrt{10 (x - 1)})}{2}$

Factor $- 1$ out of $- 1 (6) - (\sqrt{10 (x - 1)})$ .

$y = \frac{- 1 (6 + \sqrt{10 (x - 1)})}{2}$

Rewrite $- 1 (6 + \sqrt{10 (x - 1)})$ as $- (6 + \sqrt{10 (x - 1)})$ .

$y = \frac{- (6 + \sqrt{10 (x - 1)})}{2}$

Move the negative in front of the fraction.

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

The final answer is the combination of both solutions.

$y = - \frac{6 - \sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 - \sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

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 $\frac{6 - \sqrt{10 (x - 1)}}{2}$ into two fractions.

$y = - (\frac{6}{2} + \frac{- \sqrt{10 (x - 1)}}{2})$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

Step 4

Simplify each term.

Divide $6$ by $2$ .

$y = - (3 + \frac{- \sqrt{10 (x - 1)}}{2})$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

Move the negative in front of the fraction.

$y = - (3 - \frac{\sqrt{10 (x - 1)}}{2})$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

$y = - (3 - \frac{\sqrt{10 (x - 1)}}{2})$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

Step 5

Simplify by multiplying through.

Apply the distributive property.

$y = - 1 \cdot 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

Multiply $- 1$ by $3$ .

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - \frac{6 + \sqrt{10 (x - 1)}}{2}$

Step 6

Split the fraction $\frac{6 + \sqrt{10 (x - 1)}}{2}$ into two fractions.

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - (\frac{6}{2} + \frac{\sqrt{10 (x - 1)}}{2})$

Step 7

Divide $6$ by $2$ .

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - (3 + \frac{\sqrt{10 (x - 1)}}{2})$

Step 8

Apply the distributive property.

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - 1 \cdot 3 - \frac{\sqrt{10 (x - 1)}}{2}$

Step 9

Multiply $- 1$ by $3$ .

$y = - 3 + \frac{\sqrt{10 (x - 1)}}{2}$

$y = - 3 - \frac{\sqrt{10 (x - 1)}}{2}$

Step 10

Reorder terms.

$y = - 3 + \frac{1}{2} \cdot \sqrt{10 (x - 1)}$

$y = - 3 - (\frac{1}{2} \cdot \sqrt{10 (x - 1)})$

Step 11

Remove parentheses.

$y = - 3 + \frac{1}{2} \cdot \sqrt{10 (x - 1)}$

$y = - 3 - \frac{1}{2} \cdot \sqrt{10 (x - 1)}$

Step 12

Write in Standard Form 2y^2-5x+12y+23=0

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