Algebra Examples

$2 x^{2} - 4 y = - 6 x - 6 y$

Step 1

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

$2 x^{2} - 4 y + 6 x = - 6 y$

Step 2

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

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Step 2.1

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

$2 x^{2} - 4 y + 6 x + 6 y = 0$

Step 2.2

Add $- 4 y$ and $6 y$ .

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

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 $a = 2$ , $b = 6$ , and $c = 2 y$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (2 \cdot (2 y))}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 y)}}{2 \cdot 2}$

Step 5.1.2

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

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot (2 y)}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 16 y}}{2 \cdot 2}$

Step 5.1.3

Factor $4$ out of $36 - 16 y$ .

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Step 5.1.3.1

Factor $4$ out of $36$ .

$x = \frac{- 6 \pm \sqrt{4 (9) - 16 y}}{2 \cdot 2}$

Step 5.1.3.2

Factor $4$ out of $- 16 y$ .

$x = \frac{- 6 \pm \sqrt{4 (9) + 4 (- 4 y)}}{2 \cdot 2}$

Step 5.1.3.3

Factor $4$ out of $4 (9) + 4 (- 4 y)$ .

$x = \frac{- 6 \pm \sqrt{4 (9 - 4 y)}}{2 \cdot 2}$

Step 5.1.4

Rewrite $4 (9 - 4 y)$ as $2^{2} (3^{2} - 4 y)$ .

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Step 5.1.4.1

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

$x = \frac{- 6 \pm \sqrt{2^{2} (9 - 4 y)}}{2 \cdot 2}$

Step 5.1.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{3^{2} - 4 y}}{2 \cdot 2}$

Step 5.1.6

Raise $3$ to the power of $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$

Step 5.3

Simplify $\frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 y}}{2}$

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 y)}}{2 \cdot 2}$

Step 6.1.2

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

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot (2 y)}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 16 y}}{2 \cdot 2}$

Step 6.1.3

Factor $4$ out of $36 - 16 y$ .

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Step 6.1.3.1

Factor $4$ out of $36$ .

$x = \frac{- 6 \pm \sqrt{4 (9) - 16 y}}{2 \cdot 2}$

Step 6.1.3.2

Factor $4$ out of $- 16 y$ .

$x = \frac{- 6 \pm \sqrt{4 (9) + 4 (- 4 y)}}{2 \cdot 2}$

Step 6.1.3.3

Factor $4$ out of $4 (9) + 4 (- 4 y)$ .

$x = \frac{- 6 \pm \sqrt{4 (9 - 4 y)}}{2 \cdot 2}$

Step 6.1.4

Rewrite $4 (9 - 4 y)$ as $2^{2} (3^{2} - 4 y)$ .

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Step 6.1.4.1

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

$x = \frac{- 6 \pm \sqrt{2^{2} (9 - 4 y)}}{2 \cdot 2}$

Step 6.1.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{3^{2} - 4 y}}{2 \cdot 2}$

Step 6.1.6

Raise $3$ to the power of $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$

Step 6.3

Simplify $\frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 y}}{2}$

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{9 - 4 y}}{2}$

Step 6.5

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{9 - 4 y}}{2}$

Step 6.6

Factor $- 1$ out of $\sqrt{9 - 4 y}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{9 - 4 y})}{2}$

Step 6.7

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{9 - 4 y})$ .

$x = \frac{- 1 (3 - \sqrt{9 - 4 y})}{2}$

Step 6.8

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{9 - 4 y}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot (2 y)}}{2 \cdot 2}$

Step 7.1.2

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

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Step 7.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8 \cdot (2 y)}}{2 \cdot 2}$

Step 7.1.2.2

Multiply $- 8$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 16 y}}{2 \cdot 2}$

Step 7.1.3

Factor $4$ out of $36 - 16 y$ .

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Step 7.1.3.1

Factor $4$ out of $36$ .

$x = \frac{- 6 \pm \sqrt{4 (9) - 16 y}}{2 \cdot 2}$

Step 7.1.3.2

Factor $4$ out of $- 16 y$ .

$x = \frac{- 6 \pm \sqrt{4 (9) + 4 (- 4 y)}}{2 \cdot 2}$

Step 7.1.3.3

Factor $4$ out of $4 (9) + 4 (- 4 y)$ .

$x = \frac{- 6 \pm \sqrt{4 (9 - 4 y)}}{2 \cdot 2}$

Step 7.1.4

Rewrite $4 (9 - 4 y)$ as $2^{2} (3^{2} - 4 y)$ .

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Step 7.1.4.1

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

$x = \frac{- 6 \pm \sqrt{2^{2} (9 - 4 y)}}{2 \cdot 2}$

Step 7.1.4.2

Rewrite $9$ as $3^{2}$ .

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

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{3^{2} - 4 y}}{2 \cdot 2}$

Step 7.1.6

Raise $3$ to the power of $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{2 \cdot 2}$

Step 7.2

Multiply $2$ by $2$ .

$x = \frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$

Step 7.3

Simplify $\frac{- 6 \pm 2 \sqrt{9 - 4 y}}{4}$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 y}}{2}$

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{9 - 4 y}}{2}$

Step 7.5

Factor $- 1$ out of $- 3 - \sqrt{9 - 4 y}$ .

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Step 7.5.1

Reorder $9$ and $- 4 y$ .

$x = \frac{- 3 - \sqrt{- 4 y + 9}}{2}$

Step 7.5.2

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{- 4 y + 9}}{2}$

Step 7.5.3

Factor $- 1$ out of $- \sqrt{- 4 y + 9}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{- 4 y + 9})}{2}$

Step 7.5.4

Factor $- 1$ out of $- 1 (3) - (\sqrt{- 4 y + 9})$ .

$x = \frac{- 1 (3 + \sqrt{- 4 y + 9})}{2}$

Step 7.5.5

Rewrite $- 1 (3 + \sqrt{- 4 y + 9})$ as $- (3 + \sqrt{- 4 y + 9})$ .

$x = \frac{- (3 + \sqrt{- 4 y + 9})}{2}$

Step 7.6

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{- 4 y + 9}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{9 - 4 y}}{2}$

$x = - \frac{3 + \sqrt{- 4 y + 9}}{2}$