Algebra Examples

$- 6 x^{2} - 5 y = 4 x + 3 y$

Step 1

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

$- 6 x^{2} - 5 y - 4 x = 3 y$

Step 2

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

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Step 2.1

Subtract $3 y$ from both sides of the equation.

$- 6 x^{2} - 5 y - 4 x - 3 y = 0$

Step 2.2

Subtract $3 y$ from $- 5 y$ .

$- 6 x^{2} - 4 x - 8 y = 0$

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 $a = - 6$ , $b = - 4$ , and $c = - 8 y$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 6 \cdot - 8 y$ .

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Step 5.1.1.1

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 6 \cdot (- 8 y)}}{2 \cdot - 6}$

Step 5.1.1.2

Factor $- 4$ out of $- 4 \cdot - 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 6 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 5.1.2

Factor $2$ out of $- 4 - 6 \cdot - 8 y$ .

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Step 5.1.2.1

Factor $2$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 5.1.2.2

Factor $2$ out of $- 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) + 2 (- 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 5.1.2.3

Factor $2$ out of $2 (- 2) + 2 (- 3 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 - 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 5.1.3

Multiply $- 3$ by $- 8$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 + 24 y))}}{2 \cdot - 6}$

Step 5.1.4

Factor $2$ out of $- 2 + 24 y$ .

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Step 5.1.4.1

Factor $2$ out of $- 2$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 24 y))}}{2 \cdot - 6}$

Step 5.1.4.2

Factor $2$ out of $24 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 2 (12 y)))}}{2 \cdot - 6}$

Step 5.1.4.3

Factor $2$ out of $2 (- 1) + 2 (12 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

$x = \frac{4 \pm \sqrt{- 4 (2 \cdot (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

Step 5.1.5

Multiply $2$ by $2$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 5.1.6

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{- 16 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 5.1.7

Rewrite $- 16 (- 1 + 12 y)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))$ .

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Step 5.1.7.1

Factor $16$ out of $- 16$ .

$x = \frac{4 \pm \sqrt{16 (- 1) (- 1 + 12 y)}}{2 \cdot - 6}$

Step 5.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 5.1.7.3

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

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 5.1.7.4

Add parentheses.

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm 2^{2} \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 5.1.9

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

$x = \frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 5.2

Multiply $2$ by $- 6$ .

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

Step 5.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{- 12}$ .

$x = \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{- 3}$

Step 5.4

Move the negative in front of the fraction.

$x = - \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{3}$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 6 \cdot - 8 y$ .

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Step 6.1.1.1

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 6 \cdot (- 8 y)}}{2 \cdot - 6}$

Step 6.1.1.2

Factor $- 4$ out of $- 4 \cdot - 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 6.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 6 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 6.1.2

Factor $2$ out of $- 4 - 6 \cdot - 8 y$ .

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Step 6.1.2.1

Factor $2$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 6.1.2.2

Factor $2$ out of $- 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) + 2 (- 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 6.1.2.3

Factor $2$ out of $2 (- 2) + 2 (- 3 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 - 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 6.1.3

Multiply $- 3$ by $- 8$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 + 24 y))}}{2 \cdot - 6}$

Step 6.1.4

Factor $2$ out of $- 2 + 24 y$ .

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Step 6.1.4.1

Factor $2$ out of $- 2$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 24 y))}}{2 \cdot - 6}$

Step 6.1.4.2

Factor $2$ out of $24 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 2 (12 y)))}}{2 \cdot - 6}$

Step 6.1.4.3

Factor $2$ out of $2 (- 1) + 2 (12 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

$x = \frac{4 \pm \sqrt{- 4 (2 \cdot (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

Step 6.1.5

Multiply $2$ by $2$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 6.1.6

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{- 16 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 6.1.7

Rewrite $- 16 (- 1 + 12 y)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))$ .

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Step 6.1.7.1

Factor $16$ out of $- 16$ .

$x = \frac{4 \pm \sqrt{16 (- 1) (- 1 + 12 y)}}{2 \cdot - 6}$

Step 6.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 6.1.7.3

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

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 6.1.7.4

Add parentheses.

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm 2^{2} \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 6.1.9

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

$x = \frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 6.2

Multiply $2$ by $- 6$ .

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

Step 6.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{- 12}$ .

$x = \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{- 3}$

Step 6.4

Move the negative in front of the fraction.

$x = - \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{3}$

Step 6.5

Change the $\pm$ to $+$ .

$x = - \frac{1 + \sqrt{- 1 (- 1 + 12 y)}}{3}$

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

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot - 6 \cdot - 8 y$ .

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Step 7.1.1.1

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

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 \cdot - 6 \cdot (- 8 y)}}{2 \cdot - 6}$

Step 7.1.1.2

Factor $- 4$ out of $- 4 \cdot - 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot - 4 - 4 (- 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 7.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (- 6 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 7.1.2

Factor $2$ out of $- 4 - 6 \cdot - 8 y$ .

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Step 7.1.2.1

Factor $2$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) - 6 \cdot (- 8 y))}}{2 \cdot - 6}$

Step 7.1.2.2

Factor $2$ out of $- 6 \cdot - 8 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2) + 2 (- 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 7.1.2.3

Factor $2$ out of $2 (- 2) + 2 (- 3 \cdot - 8 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 - 3 \cdot (- 8 y)))}}{2 \cdot - 6}$

Step 7.1.3

Multiply $- 3$ by $- 8$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (- 2 + 24 y))}}{2 \cdot - 6}$

Step 7.1.4

Factor $2$ out of $- 2 + 24 y$ .

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Step 7.1.4.1

Factor $2$ out of $- 2$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 24 y))}}{2 \cdot - 6}$

Step 7.1.4.2

Factor $2$ out of $24 y$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1) + 2 (12 y)))}}{2 \cdot - 6}$

Step 7.1.4.3

Factor $2$ out of $2 (- 1) + 2 (12 y)$ .

$x = \frac{4 \pm \sqrt{- 4 (2 (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

$x = \frac{4 \pm \sqrt{- 4 (2 \cdot (2 (- 1 + 12 y)))}}{2 \cdot - 6}$

Step 7.1.5

Multiply $2$ by $2$ .

$x = \frac{4 \pm \sqrt{- 4 \cdot (4 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 7.1.6

Multiply $- 4$ by $4$ .

$x = \frac{4 \pm \sqrt{- 16 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 7.1.7

Rewrite $- 16 (- 1 + 12 y)$ as ${(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))$ .

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Step 7.1.7.1

Factor $16$ out of $- 16$ .

$x = \frac{4 \pm \sqrt{16 (- 1) (- 1 + 12 y)}}{2 \cdot - 6}$

Step 7.1.7.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 7.1.7.3

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

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 7.1.7.4

Add parentheses.

$x = \frac{4 \pm \sqrt{{(2^{2})}^{2} \cdot (- 1 (- 1 + 12 y))}}{2 \cdot - 6}$

Step 7.1.8

Pull terms out from under the radical.

$x = \frac{4 \pm 2^{2} \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 7.1.9

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

$x = \frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{2 \cdot - 6}$

Step 7.2

Multiply $2$ by $- 6$ .

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

Step 7.3

Simplify $\frac{4 \pm 4 \sqrt{- 1 (- 1 + 12 y)}}{- 12}$ .

$x = \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{- 3}$

Step 7.4

Move the negative in front of the fraction.

$x = - \frac{1 \pm \sqrt{- 1 (- 1 + 12 y)}}{3}$

Step 7.5

Change the $\pm$ to $-$ .

$x = - \frac{1 - \sqrt{- 1 (- 1 + 12 y)}}{3}$

Step 8

The final answer is the combination of both solutions.

$x = - \frac{1 + \sqrt{- 1 (- 1 + 12 y)}}{3}$

$x = - \frac{1 - \sqrt{- 1 (- 1 + 12 y)}}{3}$