Write as a Function of y 2y^2-3x^2=8x

$2 y^{2} - 3 x^{2} = 8 x$

Step 1

Subtract $8 x$ from both sides of the equation.

$2 y^{2} - 3 x^{2} - 8 x = 0$

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 $a = - 3$ , $b = - 8$ , and $c = 2 y^{2}$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{8 \pm \sqrt{64 + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $64 + 24 y^{2}$ .

Factor $8$ out of $64$ .

$x = \frac{8 \pm \sqrt{8 (8) + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $24 y^{2}$ .

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

Factor $8$ out of $8 (8) + 8 (3 y^{2})$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$

Simplify $\frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$ .

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

Move the negative in front of the fraction.

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

Step 5

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{8 \pm \sqrt{64 + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $64 + 24 y^{2}$ .

Factor $8$ out of $64$ .

$x = \frac{8 \pm \sqrt{8 (8) + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $24 y^{2}$ .

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

Factor $8$ out of $8 (8) + 8 (3 y^{2})$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$

Simplify $\frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $+$ .

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

Step 6

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

Simplify the numerator.

Raise $- 8$ to the power of $2$ .

$x = \frac{8 \pm \sqrt{64 - 4 \cdot - 3 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot 2$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{8 \pm \sqrt{64 + 12 \cdot (2 y^{2})}}{2 \cdot - 3}$

Multiply $12$ by $2$ .

$x = \frac{8 \pm \sqrt{64 + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $64 + 24 y^{2}$ .

Factor $8$ out of $64$ .

$x = \frac{8 \pm \sqrt{8 (8) + 24 y^{2}}}{2 \cdot - 3}$

Factor $8$ out of $24 y^{2}$ .

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

Factor $8$ out of $8 (8) + 8 (3 y^{2})$ .

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

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

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

Multiply $2$ by $- 3$ .

$x = \frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$

Simplify $\frac{8 \pm 2 \sqrt{2 (8 + 3 y^{2})}}{- 6}$ .

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

Move the negative in front of the fraction.

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

Change the $\pm$ to $-$ .

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

Step 7

The final answer is the combination of both solutions.

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

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

Write as a Function of y 2y^2-3x^2=8x

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