Solve Using the Quadratic Formula 2x^2+3=6y

$2 x^{2} + 3 = 6 y$

Subtract $6 y$ from both sides of the equation.

$2 x^{2} + 3 - 6 y = 0$

Move all terms not containing $x$ to the right side of the equation.

Subtract $3$ from both sides of the equation.

$2 x^{2} - 6 y = - 3$

Add $6 y$ to both sides of the equation.

$2 x^{2} = - 3 + 6 y$

Divide each term in $2 x^{2} = - 3 + 6 y$ by $2$ and simplify.

Divide each term in $2 x^{2} = - 3 + 6 y$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{- 3}{2} + \frac{6 y}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{- 3}{2} + \frac{6 y}{2}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 3}{2} + \frac{6 y}{2}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x^{2} = - \frac{3}{2} + \frac{6 y}{2}$

Cancel the common factor of $6$ and $2$ .

Factor $2$ out of $6 y$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

$x^{2} = - \frac{3}{2} + \frac{3 y}{1}$

Divide $3 y$ by $1$ .

$x^{2} = - \frac{3}{2} + 3 y$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\pm \sqrt{- \frac{3}{2} + 3 y}$ .

To write $3 y$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Simplify terms.

Combine $3 y$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $2$ by $3$ .

$x = \pm \sqrt{\frac{- 3 + 6 y}{2}}$

Factor $3$ out of $- 3 + 6 y$ .

Factor $3$ out of $- 3$ .

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

Factor $3$ out of $6 y$ .

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

Factor $3$ out of $3 (- 1) + 3 (2 y)$ .

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

Rewrite $\sqrt{\frac{3 (- 1 + 2 y)}{2}}$ as $\frac{\sqrt{3 (- 1 + 2 y)}}{\sqrt{2}}$ .

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

Multiply $\frac{\sqrt{3 (- 1 + 2 y)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{3 (- 1 + 2 y)}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $2$ by $3$ .

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

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

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

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

Solve Using the Quadratic Formula 2x^2+3=6y

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