Trigonometry Examples

$17 x^{2} - 12 x y + 8 y^{2} - 68 x + 24 y - 12 = 0$

Step 1

Solve the equation for $y$ .

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Step 1.1

Use the quadratic formula to find the solutions.

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

Step 1.2

Substitute the values $a = 8$ , $b = - 12 x + 24$ , and $c = 17 x^{2} - 68 x - 12$ into the quadratic formula and solve for $y$ .

$\frac{- (- 12 x + 24) \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot (8 \cdot (17 x^{2} - 68 x - 12))}}{2 \cdot 8}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Apply the distributive property.

$y = \frac{- (- 12 x) - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.3.1.2

Multiply $- 12$ by $- 1$ .

$y = \frac{12 x - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.3.1.3

Multiply $- 1$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.3.1.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot (8 \cdot (17 x^{2} - 68 x - 12))}}{2 \cdot 8}$

Step 1.3.1.5

Let $u = 8 \cdot (17 x^{2} - 68 x - 12)$ . Substitute $u$ for all occurrences of $8 \cdot (17 x^{2} - 68 x - 12)$ .

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Step 1.3.1.5.1

Rewrite ${(- 12 x + 24)}^{2}$ as $(- 12 x + 24) (- 12 x + 24)$ .

$y = \frac{12 x - 24 \pm \sqrt{(- 12 x + 24) (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.2

Expand $(- 12 x + 24) (- 12 x + 24)$ using the FOIL Method.

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Step 1.3.1.5.2.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x + 24) + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.2.2

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.2.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3

Simplify and combine like terms.

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Step 1.3.1.5.3.1

Simplify each term.

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Step 1.3.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x \cdot x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.5.3.1.2.1

Move $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 (x \cdot x)) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x^{2}) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.3

Multiply $- 12$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.4

Multiply $24$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.5

Multiply $- 12$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.1.6

Multiply $24$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 576 - 4 \cdot u}}{2 \cdot 8}$

Step 1.3.1.5.3.2

Subtract $288 x$ from $- 288 x$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 \cdot u}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.3.1.6

Factor $4$ out of $144 x^{2} - 576 x + 576 - 4 u$ .

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Step 1.3.1.6.1

Factor $4$ out of $144 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.3.1.6.2

Factor $4$ out of $- 576 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 576 - 4 u}}{2 \cdot 8}$

Step 1.3.1.6.3

Factor $4$ out of $576$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) - 4 u}}{2 \cdot 8}$

Step 1.3.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.3.1.6.5

Factor $4$ out of $4 (36 x^{2}) + 4 (- 144 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.3.1.6.6

Factor $4$ out of $4 (36 x^{2} - 144 x) + 4 (144)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144) + 4 (- u)}}{2 \cdot 8}$

Step 1.3.1.6.7

Factor $4$ out of $4 (36 x^{2} - 144 x + 144) + 4 (- u)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - u)}}{2 \cdot 8}$

Step 1.3.1.7

Replace all occurrences of $u$ with $8 \cdot (17 x^{2} - 68 x - 12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 \cdot (17 x^{2} - 68 x - 12)))}}{2 \cdot 8}$

Step 1.3.1.8

Simplify.

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Step 1.3.1.8.1

Simplify each term.

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Step 1.3.1.8.1.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 (17 x^{2}) + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.3.1.8.1.2

Simplify.

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Step 1.3.1.8.1.2.1

Multiply $17$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.3.1.8.1.2.2

Multiply $- 68$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.3.1.8.1.2.3

Multiply $8$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x - 96))}}{2 \cdot 8}$

Step 1.3.1.8.1.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2}) - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.3.1.8.1.4

Simplify.

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Step 1.3.1.8.1.4.1

Multiply $136$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.3.1.8.1.4.2

Multiply $- 544$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.3.1.8.1.4.3

Multiply $- 1$ by $- 96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.3.1.8.2

Subtract $136 x^{2}$ from $36 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} - 144 x + 144 + 544 x + 96)}}{2 \cdot 8}$

Step 1.3.1.8.3

Add $- 144 x$ and $544 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 144 + 96)}}{2 \cdot 8}$

Step 1.3.1.8.4

Add $144$ and $96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 240)}}{2 \cdot 8}$

Step 1.3.1.9

Factor $20$ out of $- 100 x^{2} + 400 x + 240$ .

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Step 1.3.1.9.1

Factor $20$ out of $- 100 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 400 x + 240)}}{2 \cdot 8}$

Step 1.3.1.9.2

Factor $20$ out of $400 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 240)}}{2 \cdot 8}$

Step 1.3.1.9.3

Factor $20$ out of $240$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.3.1.9.4

Factor $20$ out of $20 (- 5 x^{2}) + 20 (20 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.3.1.9.5

Factor $20$ out of $20 (- 5 x^{2} + 20 x) + 20 (12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{4 \cdot (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.3.1.10

Multiply $4$ by $20$ .

$y = \frac{12 x - 24 \pm \sqrt{80 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.3.1.11

Rewrite $80 (- 5 x^{2} + 20 x + 12)$ as ${(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))$ .

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Step 1.3.1.11.1

Factor $16$ out of $80$ .

$y = \frac{12 x - 24 \pm \sqrt{16 (5) (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.3.1.11.2

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

$y = \frac{12 x - 24 \pm \sqrt{4^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.3.1.11.3

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

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.3.1.11.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.3.1.12

Pull terms out from under the radical.

$y = \frac{12 x - 24 \pm 2^{2} \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.3.1.13

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

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.3.2

Multiply $2$ by $8$ .

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4

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

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Step 1.4.1

Simplify the numerator.

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Step 1.4.1.1

Apply the distributive property.

$y = \frac{- (- 12 x) - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.4.1.2

Multiply $- 12$ by $- 1$ .

$y = \frac{12 x - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.4.1.3

Multiply $- 1$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.4.1.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot (8 \cdot (17 x^{2} - 68 x - 12))}}{2 \cdot 8}$

Step 1.4.1.5

Let $u = 8 \cdot (17 x^{2} - 68 x - 12)$ . Substitute $u$ for all occurrences of $8 \cdot (17 x^{2} - 68 x - 12)$ .

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Step 1.4.1.5.1

Rewrite ${(- 12 x + 24)}^{2}$ as $(- 12 x + 24) (- 12 x + 24)$ .

$y = \frac{12 x - 24 \pm \sqrt{(- 12 x + 24) (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.2

Expand $(- 12 x + 24) (- 12 x + 24)$ using the FOIL Method.

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Step 1.4.1.5.2.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x + 24) + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.2.2

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.2.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3

Simplify and combine like terms.

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Step 1.4.1.5.3.1

Simplify each term.

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Step 1.4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x \cdot x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.5.3.1.2.1

Move $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 (x \cdot x)) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x^{2}) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.3

Multiply $- 12$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.4

Multiply $24$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.5

Multiply $- 12$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.1.6

Multiply $24$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 576 - 4 \cdot u}}{2 \cdot 8}$

Step 1.4.1.5.3.2

Subtract $288 x$ from $- 288 x$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 \cdot u}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.4.1.6

Factor $4$ out of $144 x^{2} - 576 x + 576 - 4 u$ .

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Step 1.4.1.6.1

Factor $4$ out of $144 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.4.1.6.2

Factor $4$ out of $- 576 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 576 - 4 u}}{2 \cdot 8}$

Step 1.4.1.6.3

Factor $4$ out of $576$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) - 4 u}}{2 \cdot 8}$

Step 1.4.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.4.1.6.5

Factor $4$ out of $4 (36 x^{2}) + 4 (- 144 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.4.1.6.6

Factor $4$ out of $4 (36 x^{2} - 144 x) + 4 (144)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144) + 4 (- u)}}{2 \cdot 8}$

Step 1.4.1.6.7

Factor $4$ out of $4 (36 x^{2} - 144 x + 144) + 4 (- u)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - u)}}{2 \cdot 8}$

Step 1.4.1.7

Replace all occurrences of $u$ with $8 \cdot (17 x^{2} - 68 x - 12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 \cdot (17 x^{2} - 68 x - 12)))}}{2 \cdot 8}$

Step 1.4.1.8

Simplify.

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Step 1.4.1.8.1

Simplify each term.

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Step 1.4.1.8.1.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 (17 x^{2}) + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.4.1.8.1.2

Simplify.

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Step 1.4.1.8.1.2.1

Multiply $17$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.4.1.8.1.2.2

Multiply $- 68$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.4.1.8.1.2.3

Multiply $8$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x - 96))}}{2 \cdot 8}$

Step 1.4.1.8.1.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2}) - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.4.1.8.1.4

Simplify.

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Step 1.4.1.8.1.4.1

Multiply $136$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.4.1.8.1.4.2

Multiply $- 544$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.4.1.8.1.4.3

Multiply $- 1$ by $- 96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.4.1.8.2

Subtract $136 x^{2}$ from $36 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} - 144 x + 144 + 544 x + 96)}}{2 \cdot 8}$

Step 1.4.1.8.3

Add $- 144 x$ and $544 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 144 + 96)}}{2 \cdot 8}$

Step 1.4.1.8.4

Add $144$ and $96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 240)}}{2 \cdot 8}$

Step 1.4.1.9

Factor $20$ out of $- 100 x^{2} + 400 x + 240$ .

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Step 1.4.1.9.1

Factor $20$ out of $- 100 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 400 x + 240)}}{2 \cdot 8}$

Step 1.4.1.9.2

Factor $20$ out of $400 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 240)}}{2 \cdot 8}$

Step 1.4.1.9.3

Factor $20$ out of $240$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.4.1.9.4

Factor $20$ out of $20 (- 5 x^{2}) + 20 (20 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.4.1.9.5

Factor $20$ out of $20 (- 5 x^{2} + 20 x) + 20 (12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{4 \cdot (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.4.1.10

Multiply $4$ by $20$ .

$y = \frac{12 x - 24 \pm \sqrt{80 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.4.1.11

Rewrite $80 (- 5 x^{2} + 20 x + 12)$ as ${(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))$ .

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Step 1.4.1.11.1

Factor $16$ out of $80$ .

$y = \frac{12 x - 24 \pm \sqrt{16 (5) (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.4.1.11.2

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

$y = \frac{12 x - 24 \pm \sqrt{4^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.4.1.11.3

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

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.4.1.11.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.4.1.12

Pull terms out from under the radical.

$y = \frac{12 x - 24 \pm 2^{2} \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.4.1.13

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

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.4.2

Multiply $2$ by $8$ .

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4.3

Change the $\pm$ to $+$ .

$y = \frac{12 x - 24 + 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4.4

Cancel the common factor of $12 x - 24 + 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}$ and $16$ .

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Step 1.4.4.1

Factor $4$ out of $12 x$ .

$y = \frac{4 (3 x) - 24 + 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4.4.2

Factor $4$ out of $- 24$ .

$y = \frac{4 (3 x) + 4 \cdot - 6 + 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4.4.3

Factor $4$ out of $4 (3 x) + 4 (- 6)$ .

$y = \frac{4 (3 x - 6) + 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.4.4.4

Factor $4$ out of $4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}$ .

$y = \frac{4 (3 x - 6) + 4 (\sqrt{5 (- 5 x^{2} + 20 x + 12)})}{16}$

Step 1.4.4.5

Factor $4$ out of $4 (3 x - 6) + 4 (\sqrt{5 (- 5 x^{2} + 20 x + 12)})$ .

$y = \frac{4 (3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{16}$

Step 1.4.4.6

Cancel the common factors.

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Step 1.4.4.6.1

Factor $4$ out of $16$ .

$y = \frac{4 (3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{4 (4)}$

Step 1.4.4.6.2

Cancel the common factor.

$y = \frac{4 (3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{4 \cdot 4}$

Step 1.4.4.6.3

Rewrite the expression.

$y = \frac{3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

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

Apply the distributive property.

$y = \frac{- (- 12 x) - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.5.1.2

Multiply $- 12$ by $- 1$ .

$y = \frac{12 x - 1 \cdot 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.5.1.3

Multiply $- 1$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot 8 \cdot (17 x^{2} - 68 x - 12)}}{2 \cdot 8}$

Step 1.5.1.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(- 12 x + 24)}^{2} - 4 \cdot (8 \cdot (17 x^{2} - 68 x - 12))}}{2 \cdot 8}$

Step 1.5.1.5

Let $u = 8 \cdot (17 x^{2} - 68 x - 12)$ . Substitute $u$ for all occurrences of $8 \cdot (17 x^{2} - 68 x - 12)$ .

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Step 1.5.1.5.1

Rewrite ${(- 12 x + 24)}^{2}$ as $(- 12 x + 24) (- 12 x + 24)$ .

$y = \frac{12 x - 24 \pm \sqrt{(- 12 x + 24) (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.2

Expand $(- 12 x + 24) (- 12 x + 24)$ using the FOIL Method.

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Step 1.5.1.5.2.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x + 24) + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.2.2

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x + 24) - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.2.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{- 12 x (- 12 x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3

Simplify and combine like terms.

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Step 1.5.1.5.3.1

Simplify each term.

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Step 1.5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x \cdot x) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.5.1.5.3.1.2.1

Move $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 (x \cdot x)) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.2.2

Multiply $x$ by $x$ .

$y = \frac{12 x - 24 \pm \sqrt{- 12 \cdot (- 12 x^{2}) - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.3

Multiply $- 12$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 12 x \cdot 24 + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.4

Multiply $24$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x + 24 (- 12 x) + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.5

Multiply $- 12$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 24 \cdot 24 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.1.6

Multiply $24$ by $24$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 288 x - 288 x + 576 - 4 \cdot u}}{2 \cdot 8}$

Step 1.5.1.5.3.2

Subtract $288 x$ from $- 288 x$ .

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 \cdot u}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{144 x^{2} - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.5.1.6

Factor $4$ out of $144 x^{2} - 576 x + 576 - 4 u$ .

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Step 1.5.1.6.1

Factor $4$ out of $144 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) - 576 x + 576 - 4 u}}{2 \cdot 8}$

Step 1.5.1.6.2

Factor $4$ out of $- 576 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 576 - 4 u}}{2 \cdot 8}$

Step 1.5.1.6.3

Factor $4$ out of $576$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) - 4 u}}{2 \cdot 8}$

Step 1.5.1.6.4

Factor $4$ out of $- 4 u$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2}) + 4 (- 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.5.1.6.5

Factor $4$ out of $4 (36 x^{2}) + 4 (- 144 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x) + 4 (144) + 4 (- u)}}{2 \cdot 8}$

Step 1.5.1.6.6

Factor $4$ out of $4 (36 x^{2} - 144 x) + 4 (144)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144) + 4 (- u)}}{2 \cdot 8}$

Step 1.5.1.6.7

Factor $4$ out of $4 (36 x^{2} - 144 x + 144) + 4 (- u)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - u)}}{2 \cdot 8}$

Step 1.5.1.7

Replace all occurrences of $u$ with $8 \cdot (17 x^{2} - 68 x - 12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 \cdot (17 x^{2} - 68 x - 12)))}}{2 \cdot 8}$

Step 1.5.1.8

Simplify.

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Step 1.5.1.8.1

Simplify each term.

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Step 1.5.1.8.1.1

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (8 (17 x^{2}) + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.5.1.8.1.2

Simplify.

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Step 1.5.1.8.1.2.1

Multiply $17$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} + 8 (- 68 x) + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.5.1.8.1.2.2

Multiply $- 68$ by $8$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x + 8 \cdot - 12))}}{2 \cdot 8}$

Step 1.5.1.8.1.2.3

Multiply $8$ by $- 12$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2} - 544 x - 96))}}{2 \cdot 8}$

Step 1.5.1.8.1.3

Apply the distributive property.

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - (136 x^{2}) - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.5.1.8.1.4

Simplify.

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Step 1.5.1.8.1.4.1

Multiply $136$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} - (- 544 x) + 96)}}{2 \cdot 8}$

Step 1.5.1.8.1.4.2

Multiply $- 544$ by $- 1$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.5.1.8.1.4.3

Multiply $- 1$ by $- 96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (36 x^{2} - 144 x + 144 - 136 x^{2} + 544 x + 96)}}{2 \cdot 8}$

Step 1.5.1.8.2

Subtract $136 x^{2}$ from $36 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} - 144 x + 144 + 544 x + 96)}}{2 \cdot 8}$

Step 1.5.1.8.3

Add $- 144 x$ and $544 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 144 + 96)}}{2 \cdot 8}$

Step 1.5.1.8.4

Add $144$ and $96$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (- 100 x^{2} + 400 x + 240)}}{2 \cdot 8}$

Step 1.5.1.9

Factor $20$ out of $- 100 x^{2} + 400 x + 240$ .

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Step 1.5.1.9.1

Factor $20$ out of $- 100 x^{2}$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 400 x + 240)}}{2 \cdot 8}$

Step 1.5.1.9.2

Factor $20$ out of $400 x$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 240)}}{2 \cdot 8}$

Step 1.5.1.9.3

Factor $20$ out of $240$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2}) + 20 (20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.5.1.9.4

Factor $20$ out of $20 (- 5 x^{2}) + 20 (20 x)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x) + 20 (12))}}{2 \cdot 8}$

Step 1.5.1.9.5

Factor $20$ out of $20 (- 5 x^{2} + 20 x) + 20 (12)$ .

$y = \frac{12 x - 24 \pm \sqrt{4 (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

$y = \frac{12 x - 24 \pm \sqrt{4 \cdot (20 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.5.1.10

Multiply $4$ by $20$ .

$y = \frac{12 x - 24 \pm \sqrt{80 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.5.1.11

Rewrite $80 (- 5 x^{2} + 20 x + 12)$ as ${(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))$ .

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Step 1.5.1.11.1

Factor $16$ out of $80$ .

$y = \frac{12 x - 24 \pm \sqrt{16 (5) (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.5.1.11.2

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

$y = \frac{12 x - 24 \pm \sqrt{4^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.5.1.11.3

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

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.5.1.11.4

Add parentheses.

$y = \frac{12 x - 24 \pm \sqrt{{(2^{2})}^{2} \cdot (5 (- 5 x^{2} + 20 x + 12))}}{2 \cdot 8}$

Step 1.5.1.12

Pull terms out from under the radical.

$y = \frac{12 x - 24 \pm 2^{2} \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.5.1.13

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

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{2 \cdot 8}$

Step 1.5.2

Multiply $2$ by $8$ .

$y = \frac{12 x - 24 \pm 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.5.3

Change the $\pm$ to $-$ .

$y = \frac{12 x - 24 - 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.5.4

Cancel the common factor of $12 x - 24 - 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}$ and $16$ .

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Step 1.5.4.1

Factor $4$ out of $12 x$ .

$y = \frac{4 (3 x) - 24 - 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.5.4.2

Factor $4$ out of $- 24$ .

$y = \frac{4 (3 x) + 4 (- 6) - 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.5.4.3

Factor $4$ out of $4 (3 x) + 4 (- 6)$ .

$y = \frac{4 (3 x - 6) - 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{16}$

Step 1.5.4.4

Factor $4$ out of $- 4 \sqrt{5 (- 5 x^{2} + 20 x + 12)}$ .

$y = \frac{4 (3 x - 6) + 4 (- \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{16}$

Step 1.5.4.5

Factor $4$ out of $4 (3 x - 6) + 4 (- \sqrt{5 (- 5 x^{2} + 20 x + 12)})$ .

$y = \frac{4 (3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{16}$

Step 1.5.4.6

Cancel the common factors.

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Step 1.5.4.6.1

Factor $4$ out of $16$ .

$y = \frac{4 (3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{4 (4)}$

Step 1.5.4.6.2

Cancel the common factor.

$y = \frac{4 (3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)})}{4 \cdot 4}$

Step 1.5.4.6.3

Rewrite the expression.

$y = \frac{3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

Step 1.6

The final answer is the combination of both solutions.

$y = \frac{3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

$y = \frac{3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

$y = \frac{3 x - 6 + \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

$y = \frac{3 x - 6 - \sqrt{5 (- 5 x^{2} + 20 x + 12)}}{4}$

Step 2

A linear equation is an equation of a straight line, which means that the degree of a linear equation must be $0$ or $1$ for each of its variables. In this case, the degrees of the variables in the equation violate the linear equation definition, which means that the equation is not a linear equation.

Not Linear