Write as a Function of y -3x^2+4y=4x+5y

$- 3 x^{2} + 4 y = 4 x + 5 y$

Step 1

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

$- 3 x^{2} + 4 y - 4 x = 5 y$

Step 2

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

Subtract $5 y$ from both sides of the equation.

$- 3 x^{2} + 4 y - 4 x - 5 y = 0$

Subtract $5 y$ from $4 y$ .

$- 3 x^{2} - 4 x - y = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = - 3$ , $b = - 4$ , and $c = - y$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 3 \cdot - 1 y$ .

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 3 \cdot (- 1 y)}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 3 \cdot - 1 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 3 \cdot - 1 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Multiply $- 3$ by $- 1$ .

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

Rewrite $- 4 (- 4 + 3 y)$ as $2^{2} \cdot (- 1 (- 4 + 3 y))$ .

Factor $4$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{4 (- 1) (- 4 + 3 y)}}{2 \cdot - 3}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Add parentheses.

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$ .

$x = \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{- 3}$

Move the negative in front of the fraction.

$x = - \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{3}$

Step 6

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 3 \cdot - 1 y$ .

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 3 \cdot (- 1 y)}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 3 \cdot - 1 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 3 \cdot - 1 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Multiply $- 3$ by $- 1$ .

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

Rewrite $- 4 (- 4 + 3 y)$ as $2^{2} \cdot (- 1 (- 4 + 3 y))$ .

Factor $4$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{4 (- 1) (- 4 + 3 y)}}{2 \cdot - 3}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Add parentheses.

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$ .

$x = \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{- 3}$

Move the negative in front of the fraction.

$x = - \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{3}$

Change the $\pm$ to $+$ .

$x = - \frac{2 + \sqrt{- 1 (- 4 + 3 y)}}{3}$

Step 7

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

Simplify the numerator.

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 3 \cdot - 1 y$ .

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 3 \cdot (- 1 y)}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 3 \cdot - 1 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 3 \cdot - 1 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 3 \cdot (- 1 y))}}{2 \cdot - 3}$

Multiply $- 3$ by $- 1$ .

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

Rewrite $- 4 (- 4 + 3 y)$ as $2^{2} \cdot (- 1 (- 4 + 3 y))$ .

Factor $4$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{4 (- 1) (- 4 + 3 y)}}{2 \cdot - 3}$

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Add parentheses.

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 4 + 3 y))}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 4 + 3 y)}}{- 6}$ .

$x = \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{- 3}$

Move the negative in front of the fraction.

$x = - \frac{2 \pm \sqrt{- 1 (- 4 + 3 y)}}{3}$

Change the $\pm$ to $-$ .

$x = - \frac{2 - \sqrt{- 1 (- 4 + 3 y)}}{3}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{2 + \sqrt{- 1 (- 4 + 3 y)}}{3}$

$x = - \frac{2 - \sqrt{- 1 (- 4 + 3 y)}}{3}$

Write as a Function of y -3x^2+4y=4x+5y

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