Algebra Examples

$x^{2} + 3 y^{2} - 4 x + 24 y = - 52$

Step 1

Solve for $y$ .

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Step 1.1

Add $52$ to both sides of the equation.

$x^{2} + 3 y^{2} - 4 x + 24 y + 52 = 0$

Step 1.2

Use the quadratic formula to find the solutions.

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

Step 1.3

Substitute the values $a = 3$ , $b = 24$ , and $c = x^{2} - 4 x + 52$ into the quadratic formula and solve for $y$ .

$\frac{- 24 \pm \sqrt{24^{2} - 4 \cdot (3 \cdot (x^{2} - 4 x + 52))}}{2 \cdot 3}$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 3 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.4.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.4.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} - 12 (- 4 x) - 12 \cdot 52}}{2 \cdot 3}$

Step 1.4.1.4

Simplify.

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Step 1.4.1.4.1

Multiply $- 4$ by $- 12$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 12 \cdot 52}}{2 \cdot 3}$

Step 1.4.1.4.2

Multiply $- 12$ by $52$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 624}}{2 \cdot 3}$

Step 1.4.1.5

Subtract $624$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 12 x^{2} + 48 x - 48}}{2 \cdot 3}$

Step 1.4.1.6

Rewrite $- 12 x^{2} + 48 x - 48$ in a factored form.

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Step 1.4.1.6.1

Factor $12$ out of $- 12 x^{2} + 48 x - 48$ .

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Step 1.4.1.6.1.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 48 x - 48}}{2 \cdot 3}$

Step 1.4.1.6.1.2

Factor $12$ out of $48 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) - 48}}{2 \cdot 3}$

Step 1.4.1.6.1.3

Factor $12$ out of $- 48$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.4.1.6.1.4

Factor $12$ out of $12 (- x^{2}) + 12 (4 x)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.4.1.6.1.5

Factor $12$ out of $12 (- x^{2} + 4 x) + 12 (- 4)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x - 4)}}{2 \cdot 3}$

Step 1.4.1.6.2

Factor by grouping.

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Step 1.4.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 1.4.1.6.2.1.1

Factor $4$ out of $4 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 (x) - 4)}}{2 \cdot 3}$

Step 1.4.1.6.2.1.2

Rewrite $4$ as $2$ plus $2$

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + (2 + 2) x - 4)}}{2 \cdot 3}$

Step 1.4.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 2 x + 2 x - 4)}}{2 \cdot 3}$

Step 1.4.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.4.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 24 \pm \sqrt{12 ((- x^{2} + 2 x) + 2 x - 4)}}{2 \cdot 3}$

Step 1.4.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 24 \pm \sqrt{12 (x (- x + 2) - 2 (- x + 2))}}{2 \cdot 3}$

Step 1.4.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{- 24 \pm \sqrt{12 ((- x + 2) (x - 2))}}{2 \cdot 3}$

$y = \frac{- 24 \pm \sqrt{12 (- x + 2) (x - 2)}}{2 \cdot 3}$

Step 1.4.1.6.3

Combine exponents.

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Step 1.4.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) + 2) (x - 2)}}{2 \cdot 3}$

Step 1.4.1.6.3.2

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) - 1 \cdot - 2) (x - 2)}}{2 \cdot 3}$

Step 1.4.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.4.1.6.3.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- 1 (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.4.1.6.3.5

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.4.1.6.3.6

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.4.1.6.3.7

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

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{1 + 1})}}{2 \cdot 3}$

Step 1.4.1.6.3.8

Add $1$ and $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{2})}}{2 \cdot 3}$

Step 1.4.1.6.3.9

Multiply $12$ by $- 1$ .

$y = \frac{- 24 \pm \sqrt{- 12 {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.4.1.7

Rewrite $- 12 {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2} \cdot - 3$ .

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Step 1.4.1.7.1

Factor $4$ out of $- 12$ .

$y = \frac{- 24 \pm \sqrt{4 (- 3) {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.4.1.7.2

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

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

Step 1.4.1.7.3

Move $- 3$ .

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

Step 1.4.1.7.4

Rewrite $2^{2} {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2}$ .

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

Step 1.4.1.8

Pull terms out from under the radical.

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

Step 1.4.1.9

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

$y = \frac{- 24 \pm 2 (x - 2) \sqrt{- 1 \cdot 3}}{2 \cdot 3}$

Step 1.4.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{- 24 \pm 2 (x - 2) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.4.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 1.4.1.12

Apply the distributive property.

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

Step 1.4.1.13

Multiply $2$ by $- 2$ .

$y = \frac{- 24 \pm (2 x - 4) (i \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.4.1.14

Apply the distributive property.

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 3}$

Step 1.4.2

Multiply $2$ by $3$ .

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.5

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

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Step 1.5.1

Simplify the numerator.

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Step 1.5.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 3 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.5.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.5.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} - 12 (- 4 x) - 12 \cdot 52}}{2 \cdot 3}$

Step 1.5.1.4

Simplify.

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Step 1.5.1.4.1

Multiply $- 4$ by $- 12$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 12 \cdot 52}}{2 \cdot 3}$

Step 1.5.1.4.2

Multiply $- 12$ by $52$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 624}}{2 \cdot 3}$

Step 1.5.1.5

Subtract $624$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 12 x^{2} + 48 x - 48}}{2 \cdot 3}$

Step 1.5.1.6

Rewrite $- 12 x^{2} + 48 x - 48$ in a factored form.

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Step 1.5.1.6.1

Factor $12$ out of $- 12 x^{2} + 48 x - 48$ .

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Step 1.5.1.6.1.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 48 x - 48}}{2 \cdot 3}$

Step 1.5.1.6.1.2

Factor $12$ out of $48 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) - 48}}{2 \cdot 3}$

Step 1.5.1.6.1.3

Factor $12$ out of $- 48$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.5.1.6.1.4

Factor $12$ out of $12 (- x^{2}) + 12 (4 x)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.5.1.6.1.5

Factor $12$ out of $12 (- x^{2} + 4 x) + 12 (- 4)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x - 4)}}{2 \cdot 3}$

Step 1.5.1.6.2

Factor by grouping.

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Step 1.5.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 1.5.1.6.2.1.1

Factor $4$ out of $4 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 (x) - 4)}}{2 \cdot 3}$

Step 1.5.1.6.2.1.2

Rewrite $4$ as $2$ plus $2$

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + (2 + 2) x - 4)}}{2 \cdot 3}$

Step 1.5.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 2 x + 2 x - 4)}}{2 \cdot 3}$

Step 1.5.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.5.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 24 \pm \sqrt{12 ((- x^{2} + 2 x) + 2 x - 4)}}{2 \cdot 3}$

Step 1.5.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 24 \pm \sqrt{12 (x (- x + 2) - 2 (- x + 2))}}{2 \cdot 3}$

Step 1.5.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{- 24 \pm \sqrt{12 ((- x + 2) (x - 2))}}{2 \cdot 3}$

$y = \frac{- 24 \pm \sqrt{12 (- x + 2) (x - 2)}}{2 \cdot 3}$

Step 1.5.1.6.3

Combine exponents.

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Step 1.5.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) + 2) (x - 2)}}{2 \cdot 3}$

Step 1.5.1.6.3.2

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) - 1 \cdot - 2) (x - 2)}}{2 \cdot 3}$

Step 1.5.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.5.1.6.3.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- 1 (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.5.1.6.3.5

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.5.1.6.3.6

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.5.1.6.3.7

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

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{1 + 1})}}{2 \cdot 3}$

Step 1.5.1.6.3.8

Add $1$ and $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{2})}}{2 \cdot 3}$

Step 1.5.1.6.3.9

Multiply $12$ by $- 1$ .

$y = \frac{- 24 \pm \sqrt{- 12 {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.5.1.7

Rewrite $- 12 {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2} \cdot - 3$ .

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Step 1.5.1.7.1

Factor $4$ out of $- 12$ .

$y = \frac{- 24 \pm \sqrt{4 (- 3) {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.5.1.7.2

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

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

Step 1.5.1.7.3

Move $- 3$ .

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

Step 1.5.1.7.4

Rewrite $2^{2} {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2}$ .

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

Step 1.5.1.8

Pull terms out from under the radical.

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

Step 1.5.1.9

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

$y = \frac{- 24 \pm 2 (x - 2) \sqrt{- 1 \cdot 3}}{2 \cdot 3}$

Step 1.5.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{- 24 \pm 2 (x - 2) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.5.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 1.5.1.12

Apply the distributive property.

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

Step 1.5.1.13

Multiply $2$ by $- 2$ .

$y = \frac{- 24 \pm (2 x - 4) (i \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.5.1.14

Apply the distributive property.

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 3}$

Step 1.5.2

Multiply $2$ by $3$ .

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.5.3

Change the $\pm$ to $+$ .

$y = \frac{- 24 + 2 x i \sqrt{3} - 4 i \sqrt{3}}{6}$

Step 1.5.4

Cancel the common factor of $- 24 + 2 x i \sqrt{3} - 4 i \sqrt{3}$ and $6$ .

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Step 1.5.4.1

Factor $2$ out of $- 24$ .

$y = \frac{2 \cdot - 12 + 2 x i \sqrt{3} - 4 i \sqrt{3}}{6}$

Step 1.5.4.2

Factor $2$ out of $2 x i \sqrt{3}$ .

$y = \frac{2 \cdot - 12 + 2 (x i \sqrt{3}) - 4 i \sqrt{3}}{6}$

Step 1.5.4.3

Factor $2$ out of $- 4 i \sqrt{3}$ .

$y = \frac{2 \cdot - 12 + 2 (x i \sqrt{3}) + 2 (- 2 i \sqrt{3})}{6}$

Step 1.5.4.4

Factor $2$ out of $2 (x i \sqrt{3}) + 2 (- 2 i \sqrt{3})$ .

$y = \frac{2 \cdot - 12 + 2 (x i \sqrt{3} - 2 i \sqrt{3})}{6}$

Step 1.5.4.5

Factor $2$ out of $2 \cdot - 12 + 2 (x i \sqrt{3} - 2 i \sqrt{3})$ .

$y = \frac{2 \cdot (- 12 + x i \sqrt{3} - 2 i \sqrt{3})}{6}$

Step 1.5.4.6

Cancel the common factors.

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Step 1.5.4.6.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot (- 12 + x i \sqrt{3} - 2 i \sqrt{3})}{2 (3)}$

Step 1.5.4.6.2

Cancel the common factor.

$y = \frac{2 \cdot (- 12 + x i \sqrt{3} - 2 i \sqrt{3})}{2 \cdot 3}$

Step 1.5.4.6.3

Rewrite the expression.

$y = \frac{- 12 + x i \sqrt{3} - 2 i \sqrt{3}}{3}$

Step 1.5.5

Reorder terms.

$y = \frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}$

Step 1.6

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

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Step 1.6.1

Simplify the numerator.

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Step 1.6.1.1

Raise $24$ to the power of $2$ .

$y = \frac{- 24 \pm \sqrt{576 - 4 \cdot 3 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.6.1.2

Multiply $- 4$ by $3$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 \cdot (x^{2} - 4 x + 52)}}{2 \cdot 3}$

Step 1.6.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} - 12 (- 4 x) - 12 \cdot 52}}{2 \cdot 3}$

Step 1.6.1.4

Simplify.

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Step 1.6.1.4.1

Multiply $- 4$ by $- 12$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 12 \cdot 52}}{2 \cdot 3}$

Step 1.6.1.4.2

Multiply $- 12$ by $52$ .

$y = \frac{- 24 \pm \sqrt{576 - 12 x^{2} + 48 x - 624}}{2 \cdot 3}$

Step 1.6.1.5

Subtract $624$ from $576$ .

$y = \frac{- 24 \pm \sqrt{- 12 x^{2} + 48 x - 48}}{2 \cdot 3}$

Step 1.6.1.6

Rewrite $- 12 x^{2} + 48 x - 48$ in a factored form.

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Step 1.6.1.6.1

Factor $12$ out of $- 12 x^{2} + 48 x - 48$ .

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Step 1.6.1.6.1.1

Factor $12$ out of $- 12 x^{2}$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 48 x - 48}}{2 \cdot 3}$

Step 1.6.1.6.1.2

Factor $12$ out of $48 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) - 48}}{2 \cdot 3}$

Step 1.6.1.6.1.3

Factor $12$ out of $- 48$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2}) + 12 (4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.6.1.6.1.4

Factor $12$ out of $12 (- x^{2}) + 12 (4 x)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x) + 12 (- 4)}}{2 \cdot 3}$

Step 1.6.1.6.1.5

Factor $12$ out of $12 (- x^{2} + 4 x) + 12 (- 4)$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 x - 4)}}{2 \cdot 3}$

Step 1.6.1.6.2

Factor by grouping.

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Step 1.6.1.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 1.6.1.6.2.1.1

Factor $4$ out of $4 x$ .

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 4 (x) - 4)}}{2 \cdot 3}$

Step 1.6.1.6.2.1.2

Rewrite $4$ as $2$ plus $2$

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + (2 + 2) x - 4)}}{2 \cdot 3}$

Step 1.6.1.6.2.1.3

Apply the distributive property.

$y = \frac{- 24 \pm \sqrt{12 (- x^{2} + 2 x + 2 x - 4)}}{2 \cdot 3}$

Step 1.6.1.6.2.2

Factor out the greatest common factor from each group.

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Step 1.6.1.6.2.2.1

Group the first two terms and the last two terms.

$y = \frac{- 24 \pm \sqrt{12 ((- x^{2} + 2 x) + 2 x - 4)}}{2 \cdot 3}$

Step 1.6.1.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$y = \frac{- 24 \pm \sqrt{12 (x (- x + 2) - 2 (- x + 2))}}{2 \cdot 3}$

Step 1.6.1.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

$y = \frac{- 24 \pm \sqrt{12 ((- x + 2) (x - 2))}}{2 \cdot 3}$

$y = \frac{- 24 \pm \sqrt{12 (- x + 2) (x - 2)}}{2 \cdot 3}$

Step 1.6.1.6.3

Combine exponents.

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Step 1.6.1.6.3.1

Factor $- 1$ out of $- x$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) + 2) (x - 2)}}{2 \cdot 3}$

Step 1.6.1.6.3.2

Rewrite $2$ as $- 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x) - 1 \cdot - 2) (x - 2)}}{2 \cdot 3}$

Step 1.6.1.6.3.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.6.1.6.3.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$y = \frac{- 24 \pm \sqrt{12 (- 1 (x - 2)) (x - 2)}}{2 \cdot 3}$

Step 1.6.1.6.3.5

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.6.1.6.3.6

Raise $x - 2$ to the power of $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 ((x - 2) (x - 2)))}}{2 \cdot 3}$

Step 1.6.1.6.3.7

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

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{1 + 1})}}{2 \cdot 3}$

Step 1.6.1.6.3.8

Add $1$ and $1$ .

$y = \frac{- 24 \pm \sqrt{12 \cdot (- 1 {(x - 2)}^{2})}}{2 \cdot 3}$

Step 1.6.1.6.3.9

Multiply $12$ by $- 1$ .

$y = \frac{- 24 \pm \sqrt{- 12 {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.6.1.7

Rewrite $- 12 {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2} \cdot - 3$ .

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Step 1.6.1.7.1

Factor $4$ out of $- 12$ .

$y = \frac{- 24 \pm \sqrt{4 (- 3) {(x - 2)}^{2}}}{2 \cdot 3}$

Step 1.6.1.7.2

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

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

Step 1.6.1.7.3

Move $- 3$ .

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

Step 1.6.1.7.4

Rewrite $2^{2} {(x - 2)}^{2}$ as ${(2 (x - 2))}^{2}$ .

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

Step 1.6.1.8

Pull terms out from under the radical.

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

Step 1.6.1.9

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

$y = \frac{- 24 \pm 2 (x - 2) \sqrt{- 1 \cdot 3}}{2 \cdot 3}$

Step 1.6.1.10

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$y = \frac{- 24 \pm 2 (x - 2) (\sqrt{- 1} \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.6.1.11

Rewrite $\sqrt{- 1}$ as $i$ .

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

Step 1.6.1.12

Apply the distributive property.

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

Step 1.6.1.13

Multiply $2$ by $- 2$ .

$y = \frac{- 24 \pm (2 x - 4) (i \cdot \sqrt{3})}{2 \cdot 3}$

Step 1.6.1.14

Apply the distributive property.

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{2 \cdot 3}$

Step 1.6.2

Multiply $2$ by $3$ .

$y = \frac{- 24 \pm (2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.6.3

Change the $\pm$ to $-$ .

$y = \frac{- 24 - (2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.6.4

Cancel the common factor of $- 24 - (2 x i \sqrt{3} - 4 i \sqrt{3})$ and $6$ .

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Step 1.6.4.1

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{- 1 \cdot 24 - (2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.6.4.2

Factor $- 1$ out of $- 1 (24) - (2 x i \sqrt{3} - 4 i \sqrt{3})$ .

$y = \frac{- 1 (24 + 2 x i \sqrt{3} - 4 i \sqrt{3})}{6}$

Step 1.6.4.3

Factor $2$ out of $- 1 (24 + 2 x i \sqrt{3} - 4 i \sqrt{3})$ .

$y = \frac{2 (- 1 (12 + x i \sqrt{3} - 2 i \sqrt{3}))}{6}$

Step 1.6.4.4

Cancel the common factors.

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Step 1.6.4.4.1

Factor $2$ out of $6$ .

$y = \frac{2 (- 1 (12 + x i \sqrt{3} - 2 i \sqrt{3}))}{2 (3)}$

Step 1.6.4.4.2

Cancel the common factor.

$y = \frac{2 (- 1 (12 + x i \sqrt{3} - 2 i \sqrt{3}))}{2 \cdot 3}$

Step 1.6.4.4.3

Rewrite the expression.

$y = \frac{- 1 (12 + x i \sqrt{3} - 2 i \sqrt{3})}{3}$

Step 1.6.5

Reorder terms.

$y = \frac{- 1 (i x \sqrt{3} - 2 i \sqrt{3} + 12)}{3}$

Step 1.6.6

Move the negative in front of the fraction.

$y = - \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}$

Step 1.7

The final answer is the combination of both solutions.

$y = \frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}$

$y = - \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}$

$y = \frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}$

$y = - \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}$

Step 2

Divide the first expression by the second expression.

$y = \frac{\frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}}{- \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}}$

Step 3

Expand $\frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}$ .

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Step 3.1

Split the fraction $\frac{i x \sqrt{3} - 2 i \sqrt{3} - 12}{3}$ into two fractions.

$\frac{\frac{i x \sqrt{3} - 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}}$

Step 3.2

Split the fraction $\frac{i x \sqrt{3} - 2 i \sqrt{3}}{3}$ into two fractions.

$\frac{\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}}$

Step 4

Expand $- \frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}$ .

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Step 4.1

Split the fraction $\frac{i x \sqrt{3} - 2 i \sqrt{3} + 12}{3}$ into two fractions.

$\frac{\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- (\frac{i x \sqrt{3} - 2 i \sqrt{3}}{3} + \frac{12}{3})}$

Step 4.2

Split the fraction $\frac{i x \sqrt{3} - 2 i \sqrt{3}}{3}$ into two fractions.

$\frac{\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- (\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{12}{3})}$

Step 4.3

Apply the distributive property.

$\frac{\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- (\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3}) - \frac{12}{3}}$

Step 4.4

Apply the distributive property.

$\frac{\frac{i x \sqrt{3}}{3} + \frac{- 2 i \sqrt{3}}{3} + \frac{- 12}{3}}{- \frac{i x \sqrt{3}}{3} - \frac{- 2 i \sqrt{3}}{3} - \frac{12}{3}}$

Step 5

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$\frac{i x \sqrt{3}}{3}$

$\frac{- 2 i \sqrt{3}}{3}$

$\frac{12}{3}$

$\frac{i x \sqrt{3}}{3}$

$\frac{- 2 i \sqrt{3}}{3}$

$\frac{- 12}{3}$

Step 6

Divide the highest order term in the dividend $\frac{i x \sqrt{3}}{3}$ by the highest order term in divisor $- \frac{i x \sqrt{3}}{3}$ .

						-	$1$
-	$\frac{i x \sqrt{3}}{3}$	-	$\frac{- 2 i \sqrt{3}}{3}$	-	$\frac{12}{3}$		$\frac{i x \sqrt{3}}{3}$	+	$\frac{- 2 i \sqrt{3}}{3}$	+	$\frac{- 12}{3}$

Step 7

Multiply the new quotient term by the divisor.

						-	$1$
-	$\frac{i x \sqrt{3}}{3}$	-	$\frac{- 2 i \sqrt{3}}{3}$	-	$\frac{12}{3}$		$\frac{i x \sqrt{3}}{3}$	+	$\frac{- 2 i \sqrt{3}}{3}$	+	$\frac{- 12}{3}$
						+	$\frac{i x \sqrt{3}}{3}$	-	$\frac{2 i \sqrt{3}}{3}$	+	$4$

Step 8

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{i x \sqrt{3}}{3} - \frac{2 i \sqrt{3}}{3} + 4$

						-	$1$
-	$\frac{i x \sqrt{3}}{3}$	-	$\frac{- 2 i \sqrt{3}}{3}$	-	$\frac{12}{3}$		$\frac{i x \sqrt{3}}{3}$	+	$\frac{- 2 i \sqrt{3}}{3}$	+	$\frac{- 12}{3}$
						-	$\frac{i x \sqrt{3}}{3}$	+	$\frac{2 i \sqrt{3}}{3}$	-	$4$

Step 9

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

						-	$1$
-	$\frac{i x \sqrt{3}}{3}$	-	$\frac{- 2 i \sqrt{3}}{3}$	-	$\frac{12}{3}$		$\frac{i x \sqrt{3}}{3}$	+	$\frac{- 2 i \sqrt{3}}{3}$	+	$\frac{- 12}{3}$
						-	$\frac{i x \sqrt{3}}{3}$	+	$\frac{2 i \sqrt{3}}{3}$	-	$4$
										-	$8$

Step 10

The final answer is the quotient plus the remainder over the divisor.

$y = - 1 - \frac{8}{- \frac{i x \sqrt{3}}{3} - \frac{- 2 i \sqrt{3}}{3} - \frac{12}{3}}$

Step 11