Precalculus Examples

$10 x^{2} - 4 x y + 6 y^{2} - 8 x + 8 y = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 6$ , $b = - 4 x + 8$ , and $c = 10 x^{2} - 8 x$ into the quadratic formula and solve for $y$ .

$\frac{- (- 4 x + 8) \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot (6 \cdot (10 x^{2} - 8 x))}}{2 \cdot 6}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Apply the distributive property.

$y = \frac{- (- 4 x) - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 3.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 x - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 3.1.3

Multiply $- 1$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 3.1.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot (6 \cdot (10 x^{2} - 8 x))}}{2 \cdot 6}$

Step 3.1.5

Let $u = 6 \cdot (10 x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $6 \cdot (10 x^{2} - 8 x)$ .

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Step 3.1.5.1

Rewrite ${(- 4 x + 8)}^{2}$ as $(- 4 x + 8) (- 4 x + 8)$ .

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

Step 3.1.5.2

Expand $(- 4 x + 8) (- 4 x + 8)$ using the FOIL Method.

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Step 3.1.5.2.1

Apply the distributive property.

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

Step 3.1.5.2.2

Apply the distributive property.

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

Step 3.1.5.2.3

Apply the distributive property.

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

Step 3.1.5.3

Simplify and combine like terms.

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Step 3.1.5.3.1

Simplify each term.

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Step 3.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.1.5.3.1.2

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

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Step 3.1.5.3.1.2.1

Move $x$ .

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

Step 3.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.1.5.3.1.3

Multiply $- 4$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 4 x \cdot 8 + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 3.1.5.3.1.4

Multiply $8$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 3.1.5.3.1.5

Multiply $- 4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 3.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 64 - 4 \cdot u}}{2 \cdot 6}$

Step 3.1.5.3.2

Subtract $32 x$ from $- 32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 \cdot u}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 3.1.6

Factor $4$ out of $16 x^{2} - 64 x + 64 - 4 u$ .

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Step 3.1.6.1

Factor $4$ out of $16 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 3.1.6.2

Factor $4$ out of $- 64 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 64 - 4 u}}{2 \cdot 6}$

Step 3.1.6.3

Factor $4$ out of $64$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) - 4 u}}{2 \cdot 6}$

Step 3.1.6.4

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

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 3.1.6.5

Factor $4$ out of $4 (4 x^{2}) + 4 (- 16 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 3.1.6.6

Factor $4$ out of $4 (4 x^{2} - 16 x) + 4 (16)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16) + 4 (- u)}}{2 \cdot 6}$

Step 3.1.6.7

Factor $4$ out of $4 (4 x^{2} - 16 x + 16) + 4 (- u)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - u)}}{2 \cdot 6}$

Step 3.1.7

Replace all occurrences of $u$ with $6 \cdot (10 x^{2} - 8 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 \cdot (10 x^{2} - 8 x)))}}{2 \cdot 6}$

Step 3.1.8

Simplify.

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 (10 x^{2}) + 6 (- 8 x)))}}{2 \cdot 6}$

Step 3.1.8.1.2

Multiply $10$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} + 6 (- 8 x)))}}{2 \cdot 6}$

Step 3.1.8.1.3

Multiply $- 8$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} - 48 x))}}{2 \cdot 6}$

Step 3.1.8.1.4

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2}) - (- 48 x))}}{2 \cdot 6}$

Step 3.1.8.1.5

Multiply $60$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} - (- 48 x))}}{2 \cdot 6}$

Step 3.1.8.1.6

Multiply $- 48$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} + 48 x)}}{2 \cdot 6}$

Step 3.1.8.2

Subtract $60 x^{2}$ from $4 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} - 16 x + 16 + 48 x)}}{2 \cdot 6}$

Step 3.1.8.3

Add $- 16 x$ and $48 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} + 32 x + 16)}}{2 \cdot 6}$

Step 3.1.9

Factor $8$ out of $- 56 x^{2} + 32 x + 16$ .

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Step 3.1.9.1

Factor $8$ out of $- 56 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 32 x + 16)}}{2 \cdot 6}$

Step 3.1.9.2

Factor $8$ out of $32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 16)}}{2 \cdot 6}$

Step 3.1.9.3

Factor $8$ out of $16$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 8 (2))}}{2 \cdot 6}$

Step 3.1.9.4

Factor $8$ out of $8 (- 7 x^{2}) + 8 (4 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x) + 8 (2))}}{2 \cdot 6}$

Step 3.1.9.5

Factor $8$ out of $8 (- 7 x^{2} + 4 x) + 8 (2)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{4 \cdot (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 3.1.10

Multiply $4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{32 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 3.1.11

Rewrite $32 (- 7 x^{2} + 4 x + 2)$ as ${(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))$ .

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Step 3.1.11.1

Factor $16$ out of $32$ .

$y = \frac{4 x - 8 \pm \sqrt{16 (2) (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 3.1.11.2

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

$y = \frac{4 x - 8 \pm \sqrt{4^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 3.1.11.3

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

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 3.1.11.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 3.1.12

Pull terms out from under the radical.

$y = \frac{4 x - 8 \pm 2^{2} \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 3.1.13

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

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 3.2

Multiply $2$ by $6$ .

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4

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

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$y = \frac{- (- 4 x) - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 4.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 x - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 4.1.3

Multiply $- 1$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 4.1.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot (6 \cdot (10 x^{2} - 8 x))}}{2 \cdot 6}$

Step 4.1.5

Let $u = 6 \cdot (10 x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $6 \cdot (10 x^{2} - 8 x)$ .

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Step 4.1.5.1

Rewrite ${(- 4 x + 8)}^{2}$ as $(- 4 x + 8) (- 4 x + 8)$ .

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

Step 4.1.5.2

Expand $(- 4 x + 8) (- 4 x + 8)$ using the FOIL Method.

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Step 4.1.5.2.1

Apply the distributive property.

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

Step 4.1.5.2.2

Apply the distributive property.

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

Step 4.1.5.2.3

Apply the distributive property.

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

Step 4.1.5.3

Simplify and combine like terms.

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Step 4.1.5.3.1

Simplify each term.

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Step 4.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 4.1.5.3.1.2

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

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Step 4.1.5.3.1.2.1

Move $x$ .

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

Step 4.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 4.1.5.3.1.3

Multiply $- 4$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 4 x \cdot 8 + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 4.1.5.3.1.4

Multiply $8$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 4.1.5.3.1.5

Multiply $- 4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 4.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 64 - 4 \cdot u}}{2 \cdot 6}$

Step 4.1.5.3.2

Subtract $32 x$ from $- 32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 \cdot u}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 4.1.6

Factor $4$ out of $16 x^{2} - 64 x + 64 - 4 u$ .

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Step 4.1.6.1

Factor $4$ out of $16 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 4.1.6.2

Factor $4$ out of $- 64 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 64 - 4 u}}{2 \cdot 6}$

Step 4.1.6.3

Factor $4$ out of $64$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) - 4 u}}{2 \cdot 6}$

Step 4.1.6.4

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

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 4.1.6.5

Factor $4$ out of $4 (4 x^{2}) + 4 (- 16 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 4.1.6.6

Factor $4$ out of $4 (4 x^{2} - 16 x) + 4 (16)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16) + 4 (- u)}}{2 \cdot 6}$

Step 4.1.6.7

Factor $4$ out of $4 (4 x^{2} - 16 x + 16) + 4 (- u)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - u)}}{2 \cdot 6}$

Step 4.1.7

Replace all occurrences of $u$ with $6 \cdot (10 x^{2} - 8 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 \cdot (10 x^{2} - 8 x)))}}{2 \cdot 6}$

Step 4.1.8

Simplify.

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Step 4.1.8.1

Simplify each term.

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Step 4.1.8.1.1

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 (10 x^{2}) + 6 (- 8 x)))}}{2 \cdot 6}$

Step 4.1.8.1.2

Multiply $10$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} + 6 (- 8 x)))}}{2 \cdot 6}$

Step 4.1.8.1.3

Multiply $- 8$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} - 48 x))}}{2 \cdot 6}$

Step 4.1.8.1.4

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2}) - (- 48 x))}}{2 \cdot 6}$

Step 4.1.8.1.5

Multiply $60$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} - (- 48 x))}}{2 \cdot 6}$

Step 4.1.8.1.6

Multiply $- 48$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} + 48 x)}}{2 \cdot 6}$

Step 4.1.8.2

Subtract $60 x^{2}$ from $4 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} - 16 x + 16 + 48 x)}}{2 \cdot 6}$

Step 4.1.8.3

Add $- 16 x$ and $48 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} + 32 x + 16)}}{2 \cdot 6}$

Step 4.1.9

Factor $8$ out of $- 56 x^{2} + 32 x + 16$ .

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Step 4.1.9.1

Factor $8$ out of $- 56 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 32 x + 16)}}{2 \cdot 6}$

Step 4.1.9.2

Factor $8$ out of $32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 16)}}{2 \cdot 6}$

Step 4.1.9.3

Factor $8$ out of $16$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 8 (2))}}{2 \cdot 6}$

Step 4.1.9.4

Factor $8$ out of $8 (- 7 x^{2}) + 8 (4 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x) + 8 (2))}}{2 \cdot 6}$

Step 4.1.9.5

Factor $8$ out of $8 (- 7 x^{2} + 4 x) + 8 (2)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{4 \cdot (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 4.1.10

Multiply $4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{32 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 4.1.11

Rewrite $32 (- 7 x^{2} + 4 x + 2)$ as ${(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))$ .

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Step 4.1.11.1

Factor $16$ out of $32$ .

$y = \frac{4 x - 8 \pm \sqrt{16 (2) (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 4.1.11.2

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

$y = \frac{4 x - 8 \pm \sqrt{4^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 4.1.11.3

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

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 4.1.11.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 4.1.12

Pull terms out from under the radical.

$y = \frac{4 x - 8 \pm 2^{2} \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 4.1.13

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

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 4.2

Multiply $2$ by $6$ .

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4.3

Change the $\pm$ to $+$ .

$y = \frac{4 x - 8 + 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4.4

Cancel the common factor of $4 x - 8 + 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}$ and $12$ .

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Step 4.4.1

Factor $4$ out of $4 x$ .

$y = \frac{4 (x) - 8 + 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4.4.2

Factor $4$ out of $- 8$ .

$y = \frac{4 (x) + 4 \cdot - 2 + 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4.4.3

Factor $4$ out of $4 (x) + 4 (- 2)$ .

$y = \frac{4 (x - 2) + 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 4.4.4

Factor $4$ out of $4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}$ .

$y = \frac{4 (x - 2) + 4 (\sqrt{2 (- 7 x^{2} + 4 x + 2)})}{12}$

Step 4.4.5

Factor $4$ out of $4 (x - 2) + 4 (\sqrt{2 (- 7 x^{2} + 4 x + 2)})$ .

$y = \frac{4 (x - 2 + \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{12}$

Step 4.4.6

Cancel the common factors.

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Step 4.4.6.1

Factor $4$ out of $12$ .

$y = \frac{4 (x - 2 + \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{4 (3)}$

Step 4.4.6.2

Cancel the common factor.

$y = \frac{4 (x - 2 + \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{4 \cdot 3}$

Step 4.4.6.3

Rewrite the expression.

$y = \frac{x - 2 + \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{3}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$y = \frac{- (- 4 x) - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 5.1.2

Multiply $- 4$ by $- 1$ .

$y = \frac{4 x - 1 \cdot 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 5.1.3

Multiply $- 1$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot 6 \cdot (10 x^{2} - 8 x)}}{2 \cdot 6}$

Step 5.1.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(- 4 x + 8)}^{2} - 4 \cdot (6 \cdot (10 x^{2} - 8 x))}}{2 \cdot 6}$

Step 5.1.5

Let $u = 6 \cdot (10 x^{2} - 8 x)$ . Substitute $u$ for all occurrences of $6 \cdot (10 x^{2} - 8 x)$ .

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Step 5.1.5.1

Rewrite ${(- 4 x + 8)}^{2}$ as $(- 4 x + 8) (- 4 x + 8)$ .

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

Step 5.1.5.2

Expand $(- 4 x + 8) (- 4 x + 8)$ using the FOIL Method.

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Step 5.1.5.2.1

Apply the distributive property.

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

Step 5.1.5.2.2

Apply the distributive property.

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

Step 5.1.5.2.3

Apply the distributive property.

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

Step 5.1.5.3

Simplify and combine like terms.

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Step 5.1.5.3.1

Simplify each term.

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Step 5.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.5.3.1.2

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

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Step 5.1.5.3.1.2.1

Move $x$ .

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

Step 5.1.5.3.1.2.2

Multiply $x$ by $x$ .

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

Step 5.1.5.3.1.3

Multiply $- 4$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 4 x \cdot 8 + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 5.1.5.3.1.4

Multiply $8$ by $- 4$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x + 8 (- 4 x) + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 5.1.5.3.1.5

Multiply $- 4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 8 \cdot 8 - 4 \cdot u}}{2 \cdot 6}$

Step 5.1.5.3.1.6

Multiply $8$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 32 x - 32 x + 64 - 4 \cdot u}}{2 \cdot 6}$

Step 5.1.5.3.2

Subtract $32 x$ from $- 32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 \cdot u}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{16 x^{2} - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 5.1.6

Factor $4$ out of $16 x^{2} - 64 x + 64 - 4 u$ .

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Step 5.1.6.1

Factor $4$ out of $16 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) - 64 x + 64 - 4 u}}{2 \cdot 6}$

Step 5.1.6.2

Factor $4$ out of $- 64 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 64 - 4 u}}{2 \cdot 6}$

Step 5.1.6.3

Factor $4$ out of $64$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) - 4 u}}{2 \cdot 6}$

Step 5.1.6.4

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

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2}) + 4 (- 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 5.1.6.5

Factor $4$ out of $4 (4 x^{2}) + 4 (- 16 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x) + 4 (16) + 4 (- u)}}{2 \cdot 6}$

Step 5.1.6.6

Factor $4$ out of $4 (4 x^{2} - 16 x) + 4 (16)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16) + 4 (- u)}}{2 \cdot 6}$

Step 5.1.6.7

Factor $4$ out of $4 (4 x^{2} - 16 x + 16) + 4 (- u)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - u)}}{2 \cdot 6}$

Step 5.1.7

Replace all occurrences of $u$ with $6 \cdot (10 x^{2} - 8 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 \cdot (10 x^{2} - 8 x)))}}{2 \cdot 6}$

Step 5.1.8

Simplify.

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Step 5.1.8.1

Simplify each term.

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Step 5.1.8.1.1

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (6 (10 x^{2}) + 6 (- 8 x)))}}{2 \cdot 6}$

Step 5.1.8.1.2

Multiply $10$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} + 6 (- 8 x)))}}{2 \cdot 6}$

Step 5.1.8.1.3

Multiply $- 8$ by $6$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2} - 48 x))}}{2 \cdot 6}$

Step 5.1.8.1.4

Apply the distributive property.

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - (60 x^{2}) - (- 48 x))}}{2 \cdot 6}$

Step 5.1.8.1.5

Multiply $60$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} - (- 48 x))}}{2 \cdot 6}$

Step 5.1.8.1.6

Multiply $- 48$ by $- 1$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (4 x^{2} - 16 x + 16 - 60 x^{2} + 48 x)}}{2 \cdot 6}$

Step 5.1.8.2

Subtract $60 x^{2}$ from $4 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} - 16 x + 16 + 48 x)}}{2 \cdot 6}$

Step 5.1.8.3

Add $- 16 x$ and $48 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (- 56 x^{2} + 32 x + 16)}}{2 \cdot 6}$

Step 5.1.9

Factor $8$ out of $- 56 x^{2} + 32 x + 16$ .

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Step 5.1.9.1

Factor $8$ out of $- 56 x^{2}$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 32 x + 16)}}{2 \cdot 6}$

Step 5.1.9.2

Factor $8$ out of $32 x$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 16)}}{2 \cdot 6}$

Step 5.1.9.3

Factor $8$ out of $16$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2}) + 8 (4 x) + 8 (2))}}{2 \cdot 6}$

Step 5.1.9.4

Factor $8$ out of $8 (- 7 x^{2}) + 8 (4 x)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x) + 8 (2))}}{2 \cdot 6}$

Step 5.1.9.5

Factor $8$ out of $8 (- 7 x^{2} + 4 x) + 8 (2)$ .

$y = \frac{4 x - 8 \pm \sqrt{4 (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

$y = \frac{4 x - 8 \pm \sqrt{4 \cdot (8 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 5.1.10

Multiply $4$ by $8$ .

$y = \frac{4 x - 8 \pm \sqrt{32 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 5.1.11

Rewrite $32 (- 7 x^{2} + 4 x + 2)$ as ${(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))$ .

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Step 5.1.11.1

Factor $16$ out of $32$ .

$y = \frac{4 x - 8 \pm \sqrt{16 (2) (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 5.1.11.2

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

$y = \frac{4 x - 8 \pm \sqrt{4^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 5.1.11.3

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

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 5.1.11.4

Add parentheses.

$y = \frac{4 x - 8 \pm \sqrt{{(2^{2})}^{2} \cdot (2 (- 7 x^{2} + 4 x + 2))}}{2 \cdot 6}$

Step 5.1.12

Pull terms out from under the radical.

$y = \frac{4 x - 8 \pm 2^{2} \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 5.1.13

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

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{2 \cdot 6}$

Step 5.2

Multiply $2$ by $6$ .

$y = \frac{4 x - 8 \pm 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 5.3

Change the $\pm$ to $-$ .

$y = \frac{4 x - 8 - 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 5.4

Cancel the common factor of $4 x - 8 - 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}$ and $12$ .

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Step 5.4.1

Factor $4$ out of $4 x$ .

$y = \frac{4 (x) - 8 - 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 5.4.2

Factor $4$ out of $- 8$ .

$y = \frac{4 (x) + 4 (- 2) - 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 5.4.3

Factor $4$ out of $4 (x) + 4 (- 2)$ .

$y = \frac{4 (x - 2) - 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{12}$

Step 5.4.4

Factor $4$ out of $- 4 \sqrt{2 (- 7 x^{2} + 4 x + 2)}$ .

$y = \frac{4 (x - 2) + 4 (- \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{12}$

Step 5.4.5

Factor $4$ out of $4 (x - 2) + 4 (- \sqrt{2 (- 7 x^{2} + 4 x + 2)})$ .

$y = \frac{4 (x - 2 - \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{12}$

Step 5.4.6

Cancel the common factors.

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Step 5.4.6.1

Factor $4$ out of $12$ .

$y = \frac{4 (x - 2 - \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{4 (3)}$

Step 5.4.6.2

Cancel the common factor.

$y = \frac{4 (x - 2 - \sqrt{2 (- 7 x^{2} + 4 x + 2)})}{4 \cdot 3}$

Step 5.4.6.3

Rewrite the expression.

$y = \frac{x - 2 - \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{3}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{x - 2 + \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{3}$

$y = \frac{x - 2 - \sqrt{2 (- 7 x^{2} + 4 x + 2)}}{3}$