Linear Algebra Examples

$d = v (x^{2} - x + (y^{2} - y)) \cdot 22$

Step 1

Rewrite the equation as $v (x^{2} - x + y^{2} - y) \cdot 22 = d$ .

$v (x^{2} - x + y^{2} - y) \cdot 22 = d$

Step 2

Simplify $v (x^{2} - x + y^{2} - y) \cdot 22$ .

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Step 2.1

Apply the distributive property.

$(v x^{2} + v (- x) + v y^{2} + v (- y)) \cdot 22 = d$

Step 2.2

Simplify.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$(v x^{2} - v x + v y^{2} + v (- y)) \cdot 22 = d$

Step 2.2.2

Rewrite using the commutative property of multiplication.

$(v x^{2} - v x + v y^{2} - v y) \cdot 22 = d$

Step 2.3

Apply the distributive property.

$v x^{2} \cdot 22 - v x \cdot 22 + v y^{2} \cdot 22 - v y \cdot 22 = d$

Step 2.4

Simplify.

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Step 2.4.1

Move $22$ to the left of $v x^{2}$ .

$22 \cdot (v x^{2}) - v x \cdot 22 + v y^{2} \cdot 22 - v y \cdot 22 = d$

Step 2.4.2

Multiply $22$ by $- 1$ .

$22 \cdot (v x^{2}) - 22 v x + v y^{2} \cdot 22 - v y \cdot 22 = d$

Step 2.4.3

Move $22$ to the left of $v y^{2}$ .

$22 \cdot (v x^{2}) - 22 v x + 22 \cdot (v y^{2}) - v y \cdot 22 = d$

Step 2.4.4

Multiply $22$ by $- 1$ .

$22 \cdot (v x^{2}) - 22 v x + 22 \cdot (v y^{2}) - 22 v y = d$

Step 2.5

Remove parentheses.

$22 v x^{2} - 22 v x + 22 v y^{2} - 22 v y = d$

Step 3

Subtract $d$ from both sides of the equation.

$22 v x^{2} - 22 v x + 22 v y^{2} - 22 v y - d = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 22 v$ , $b = - 22 v$ , and $c = 22 v x^{2} - 22 v x - d$ into the quadratic formula and solve for $y$ .

$\frac{22 v \pm \sqrt{{(- 22 v)}^{2} - 4 \cdot (22 v \cdot (22 v x^{2} - 22 v x - d))}}{2 (22 v)}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Add parentheses.

$y = \frac{22 v \pm \sqrt{{(- 22 v)}^{2} - 4 \cdot (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d)))}}{2 \cdot (22 v)}$

Step 6.1.2

Let $u = 22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ . Substitute $u$ for all occurrences of $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

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Step 6.1.2.1

Apply the product rule to $- 22 v$ .

$y = \frac{22 v \pm \sqrt{{(- 22)}^{2} v^{2} - 4 \cdot u}}{2 \cdot (22 v)}$

Step 6.1.2.2

Raise $- 22$ to the power of $2$ .

$y = \frac{22 v \pm \sqrt{484 v^{2} - 4 u}}{2 \cdot (22 v)}$

Step 6.1.3

Factor $4$ out of $484 v^{2} - 4 u$ .

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Step 6.1.3.1

Factor $4$ out of $484 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) - 4 u}}{2 \cdot (22 v)}$

Step 6.1.3.2

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

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) + 4 (- u)}}{2 \cdot (22 v)}$

Step 6.1.3.3

Factor $4$ out of $4 (121 v^{2}) + 4 (- u)$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - u)}}{2 \cdot (22 v)}$

Step 6.1.4

Replace all occurrences of $u$ with $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))))}}{2 \cdot (22 v)}$

Step 6.1.5

Simplify each term.

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Step 6.1.5.1

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 v x^{2} - 22 v x - d))))}}{2 \cdot (22 v)}$

Step 6.1.5.2

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v (22 v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 6.1.5.3

Simplify.

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Step 6.1.5.3.1

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 6.1.5.3.2

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 6.1.5.3.3

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.4

Simplify each term.

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Step 6.1.5.4.1

Multiply $v$ by $v$ by adding the exponents.

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Step 6.1.5.4.1.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 (v \cdot v) x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.4.1.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.4.2

Multiply $v$ by $v$ by adding the exponents.

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Step 6.1.5.4.2.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 (v \cdot v) x - v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.4.2.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v^{2} x - v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.5

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.6

Simplify.

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Step 6.1.5.6.1

Multiply $22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.6.2

Multiply $- 22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.6.3

Multiply $- 1$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) - 22 (v d)))}}{2 \cdot (22 v)}$

Step 6.1.5.7

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2} - 484 v^{2} x - 22 v d))}}{2 \cdot (22 v)}$

Step 6.1.5.8

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 6.1.5.9

Simplify.

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Step 6.1.5.9.1

Multiply $484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 6.1.5.9.2

Multiply $- 484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 6.1.5.9.3

Multiply $- 22$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) + 22 (v d))}}{2 \cdot (22 v)}$

Step 6.1.5.10

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 6.1.6

Factor $11 v$ out of $121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d$ .

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Step 6.1.6.1

Factor $11 v$ out of $121 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 6.1.6.2

Factor $11 v$ out of $- 484 v^{2} x^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 6.1.6.3

Factor $11 v$ out of $484 v^{2} x$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 22 v d)}}{2 \cdot (22 v)}$

Step 6.1.6.4

Factor $11 v$ out of $22 v d$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 6.1.6.5

Factor $11 v$ out of $11 v (11 v) + 11 v (- 44 v x^{2})$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 6.1.6.6

Factor $11 v$ out of $11 v (11 v - 44 v x^{2}) + 11 v (44 v x)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 6.1.6.7

Factor $11 v$ out of $11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

$y = \frac{22 v \pm \sqrt{4 \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 6.1.7

Multiply $4$ by $11$ .

$y = \frac{22 v \pm \sqrt{44 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 6.1.8

Rewrite $44 v (11 v - 44 v x^{2} + 44 v x + 2 d)$ as $2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))$ .

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Step 6.1.8.1

Factor $4$ out of $44$ .

$y = \frac{22 v \pm \sqrt{4 (11) v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 6.1.8.2

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

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 6.1.8.3

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 (v (11 v - 44 v x^{2} + 44 v x + 2 d)))}}{2 \cdot (22 v)}$

Step 6.1.8.4

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 6.1.9

Pull terms out from under the radical.

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 6.2

Multiply $2$ by $22$ .

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$

Step 6.3

Simplify $\frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$ .

$y = \frac{11 v \pm \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

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

Add parentheses.

$y = \frac{22 v \pm \sqrt{{(- 22 v)}^{2} - 4 \cdot (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d)))}}{2 \cdot (22 v)}$

Step 7.1.2

Let $u = 22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ . Substitute $u$ for all occurrences of $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

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Step 7.1.2.1

Apply the product rule to $- 22 v$ .

$y = \frac{22 v \pm \sqrt{{(- 22)}^{2} v^{2} - 4 \cdot u}}{2 \cdot (22 v)}$

Step 7.1.2.2

Raise $- 22$ to the power of $2$ .

$y = \frac{22 v \pm \sqrt{484 v^{2} - 4 u}}{2 \cdot (22 v)}$

Step 7.1.3

Factor $4$ out of $484 v^{2} - 4 u$ .

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Step 7.1.3.1

Factor $4$ out of $484 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) - 4 u}}{2 \cdot (22 v)}$

Step 7.1.3.2

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

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) + 4 (- u)}}{2 \cdot (22 v)}$

Step 7.1.3.3

Factor $4$ out of $4 (121 v^{2}) + 4 (- u)$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - u)}}{2 \cdot (22 v)}$

Step 7.1.4

Replace all occurrences of $u$ with $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))))}}{2 \cdot (22 v)}$

Step 7.1.5

Simplify each term.

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Step 7.1.5.1

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 v x^{2} - 22 v x - d))))}}{2 \cdot (22 v)}$

Step 7.1.5.2

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v (22 v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 7.1.5.3

Simplify.

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Step 7.1.5.3.1

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 7.1.5.3.2

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 7.1.5.3.3

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.4

Simplify each term.

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Step 7.1.5.4.1

Multiply $v$ by $v$ by adding the exponents.

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Step 7.1.5.4.1.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 (v \cdot v) x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.4.1.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.4.2

Multiply $v$ by $v$ by adding the exponents.

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Step 7.1.5.4.2.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 (v \cdot v) x - v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.4.2.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v^{2} x - v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.5

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.6

Simplify.

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Step 7.1.5.6.1

Multiply $22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.6.2

Multiply $- 22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.6.3

Multiply $- 1$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) - 22 (v d)))}}{2 \cdot (22 v)}$

Step 7.1.5.7

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2} - 484 v^{2} x - 22 v d))}}{2 \cdot (22 v)}$

Step 7.1.5.8

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 7.1.5.9

Simplify.

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Step 7.1.5.9.1

Multiply $484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 7.1.5.9.2

Multiply $- 484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 7.1.5.9.3

Multiply $- 22$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) + 22 (v d))}}{2 \cdot (22 v)}$

Step 7.1.5.10

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 7.1.6

Factor $11 v$ out of $121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d$ .

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Step 7.1.6.1

Factor $11 v$ out of $121 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 7.1.6.2

Factor $11 v$ out of $- 484 v^{2} x^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 7.1.6.3

Factor $11 v$ out of $484 v^{2} x$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 22 v d)}}{2 \cdot (22 v)}$

Step 7.1.6.4

Factor $11 v$ out of $22 v d$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 7.1.6.5

Factor $11 v$ out of $11 v (11 v) + 11 v (- 44 v x^{2})$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 7.1.6.6

Factor $11 v$ out of $11 v (11 v - 44 v x^{2}) + 11 v (44 v x)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 7.1.6.7

Factor $11 v$ out of $11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

$y = \frac{22 v \pm \sqrt{4 \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 7.1.7

Multiply $4$ by $11$ .

$y = \frac{22 v \pm \sqrt{44 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 7.1.8

Rewrite $44 v (11 v - 44 v x^{2} + 44 v x + 2 d)$ as $2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))$ .

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Step 7.1.8.1

Factor $4$ out of $44$ .

$y = \frac{22 v \pm \sqrt{4 (11) v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 7.1.8.2

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

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 7.1.8.3

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 (v (11 v - 44 v x^{2} + 44 v x + 2 d)))}}{2 \cdot (22 v)}$

Step 7.1.8.4

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 7.1.9

Pull terms out from under the radical.

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 7.2

Multiply $2$ by $22$ .

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$

Step 7.3

Simplify $\frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$ .

$y = \frac{11 v \pm \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

Step 7.4

Change the $\pm$ to $+$ .

$y = \frac{11 v + \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

Step 8

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

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Add parentheses.

$y = \frac{22 v \pm \sqrt{{(- 22 v)}^{2} - 4 \cdot (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d)))}}{2 \cdot (22 v)}$

Step 8.1.2

Let $u = 22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ . Substitute $u$ for all occurrences of $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

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Step 8.1.2.1

Apply the product rule to $- 22 v$ .

$y = \frac{22 v \pm \sqrt{{(- 22)}^{2} v^{2} - 4 \cdot u}}{2 \cdot (22 v)}$

Step 8.1.2.2

Raise $- 22$ to the power of $2$ .

$y = \frac{22 v \pm \sqrt{484 v^{2} - 4 u}}{2 \cdot (22 v)}$

Step 8.1.3

Factor $4$ out of $484 v^{2} - 4 u$ .

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Step 8.1.3.1

Factor $4$ out of $484 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) - 4 u}}{2 \cdot (22 v)}$

Step 8.1.3.2

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

$y = \frac{22 v \pm \sqrt{4 (121 v^{2}) + 4 (- u)}}{2 \cdot (22 v)}$

Step 8.1.3.3

Factor $4$ out of $4 (121 v^{2}) + 4 (- u)$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - u)}}{2 \cdot (22 v)}$

Step 8.1.4

Replace all occurrences of $u$ with $22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 (v x^{2}) - 22 (v x) - d))))}}{2 \cdot (22 v)}$

Step 8.1.5

Simplify each term.

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Step 8.1.5.1

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v \cdot (22 v x^{2} - 22 v x - d))))}}{2 \cdot (22 v)}$

Step 8.1.5.2

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (v (22 v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 8.1.5.3

Simplify.

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Step 8.1.5.3.1

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) + v (- 22 v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 8.1.5.3.2

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) + v (- d))))}}{2 \cdot (22 v)}$

Step 8.1.5.3.3

Rewrite using the commutative property of multiplication.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v (v x^{2}) - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.4

Simplify each term.

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Step 8.1.5.4.1

Multiply $v$ by $v$ by adding the exponents.

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Step 8.1.5.4.1.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 (v \cdot v) x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.4.1.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v (v x) - v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.4.2

Multiply $v$ by $v$ by adding the exponents.

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Step 8.1.5.4.2.1

Move $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 (v \cdot v) x - v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.4.2.2

Multiply $v$ by $v$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2} - 22 v^{2} x - v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.5

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (22 (22 v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.6

Simplify.

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Step 8.1.5.6.1

Multiply $22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) + 22 (- 22 v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.6.2

Multiply $- 22$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) + 22 (- v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.6.3

Multiply $- 1$ by $22$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 (v^{2} x^{2}) - 484 (v^{2} x) - 22 (v d)))}}{2 \cdot (22 v)}$

Step 8.1.5.7

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2} - 484 v^{2} x - 22 v d))}}{2 \cdot (22 v)}$

Step 8.1.5.8

Apply the distributive property.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - (484 v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 8.1.5.9

Simplify.

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Step 8.1.5.9.1

Multiply $484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) - (- 484 v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 8.1.5.9.2

Multiply $- 484$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) - (- 22 v d))}}{2 \cdot (22 v)}$

Step 8.1.5.9.3

Multiply $- 22$ by $- 1$ .

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 (v^{2} x^{2}) + 484 (v^{2} x) + 22 (v d))}}{2 \cdot (22 v)}$

Step 8.1.5.10

Remove parentheses.

$y = \frac{22 v \pm \sqrt{4 (121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 8.1.6

Factor $11 v$ out of $121 v^{2} - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d$ .

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Step 8.1.6.1

Factor $11 v$ out of $121 v^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) - 484 v^{2} x^{2} + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 8.1.6.2

Factor $11 v$ out of $- 484 v^{2} x^{2}$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 484 v^{2} x + 22 v d)}}{2 \cdot (22 v)}$

Step 8.1.6.3

Factor $11 v$ out of $484 v^{2} x$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 22 v d)}}{2 \cdot (22 v)}$

Step 8.1.6.4

Factor $11 v$ out of $22 v d$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v) + 11 v (- 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 8.1.6.5

Factor $11 v$ out of $11 v (11 v) + 11 v (- 44 v x^{2})$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2}) + 11 v (44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 8.1.6.6

Factor $11 v$ out of $11 v (11 v - 44 v x^{2}) + 11 v (44 v x)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d))}}{2 \cdot (22 v)}$

Step 8.1.6.7

Factor $11 v$ out of $11 v (11 v - 44 v x^{2} + 44 v x) + 11 v (2 d)$ .

$y = \frac{22 v \pm \sqrt{4 (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

$y = \frac{22 v \pm \sqrt{4 \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 8.1.7

Multiply $4$ by $11$ .

$y = \frac{22 v \pm \sqrt{44 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 8.1.8

Rewrite $44 v (11 v - 44 v x^{2} + 44 v x + 2 d)$ as $2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))$ .

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Step 8.1.8.1

Factor $4$ out of $44$ .

$y = \frac{22 v \pm \sqrt{4 (11) v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 8.1.8.2

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

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 8.1.8.3

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 (v (11 v - 44 v x^{2} + 44 v x + 2 d)))}}{2 \cdot (22 v)}$

Step 8.1.8.4

Add parentheses.

$y = \frac{22 v \pm \sqrt{2^{2} \cdot (11 v (11 v - 44 v x^{2} + 44 v x + 2 d))}}{2 \cdot (22 v)}$

Step 8.1.9

Pull terms out from under the radical.

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{2 \cdot (22 v)}$

Step 8.2

Multiply $2$ by $22$ .

$y = \frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$

Step 8.3

Simplify $\frac{22 v \pm 2 \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{44 v}$ .

$y = \frac{11 v \pm \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

Step 8.4

Change the $\pm$ to $-$ .

$y = \frac{11 v - \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

Step 9

The final answer is the combination of both solutions.

$y = \frac{11 v + \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

$y = \frac{11 v - \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$

Step 10

Set the radicand in $\sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}$ greater than or equal to $0$ to find where the expression is defined.

$11 v (11 v - 44 v x^{2} + 44 v x + 2 d) \geq 0$

Step 11

Solve for $v$ .

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Step 11.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$v = 0$

$11 v - 44 v x^{2} + 44 v x + 2 d = 0$

Step 11.2

Set $v$ equal to $0$ .

$v = 0$

Step 11.3

Set $11 v - 44 v x^{2} + 44 v x + 2 d$ equal to $0$ and solve for $v$ .

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Step 11.3.1

Set $11 v - 44 v x^{2} + 44 v x + 2 d$ equal to $0$ .

$11 v - 44 v x^{2} + 44 v x + 2 d = 0$

Step 11.3.2

Solve $11 v - 44 v x^{2} + 44 v x + 2 d = 0$ for $v$ .

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Step 11.3.2.1

Subtract $2 d$ from both sides of the equation.

$11 v - 44 v x^{2} + 44 v x = - 2 d$

Step 11.3.2.2

Factor $11 v$ out of $11 v - 44 v x^{2} + 44 v x$ .

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Step 11.3.2.2.1

Factor $11 v$ out of $11 v$ .

$11 v (1) - 44 v x^{2} + 44 v x = - 2 d$

Step 11.3.2.2.2

Factor $11 v$ out of $- 44 v x^{2}$ .

$11 v (1) + 11 v (- 4 x^{2}) + 44 v x = - 2 d$

Step 11.3.2.2.3

Factor $11 v$ out of $44 v x$ .

$11 v (1) + 11 v (- 4 x^{2}) + 11 v (4 x) = - 2 d$

Step 11.3.2.2.4

Factor $11 v$ out of $11 v (1) + 11 v (- 4 x^{2})$ .

$11 v (1 - 4 x^{2}) + 11 v (4 x) = - 2 d$

Step 11.3.2.2.5

Factor $11 v$ out of $11 v (1 - 4 x^{2}) + 11 v (4 x)$ .

$11 v (1 - 4 x^{2} + 4 x) = - 2 d$

Step 11.3.2.3

Reorder terms.

$11 v (- 4 x^{2} + 4 x + 1) = - 2 d$

Step 11.3.2.4

Divide each term in $11 v (- 4 x^{2} + 4 x + 1) = - 2 d$ by $11 (- 4 x^{2} + 4 x + 1)$ and simplify.

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Step 11.3.2.4.1

Divide each term in $11 v (- 4 x^{2} + 4 x + 1) = - 2 d$ by $11 (- 4 x^{2} + 4 x + 1)$ .

$\frac{11 v (- 4 x^{2} + 4 x + 1)}{11 (- 4 x^{2} + 4 x + 1)} = \frac{- 2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.2

Simplify the left side.

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Step 11.3.2.4.2.1

Cancel the common factor of $11$ .

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Step 11.3.2.4.2.1.1

Cancel the common factor.

$\frac{11 v (- 4 x^{2} + 4 x + 1)}{11 (- 4 x^{2} + 4 x + 1)} = \frac{- 2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.2.1.2

Rewrite the expression.

$\frac{v (- 4 x^{2} + 4 x + 1)}{- 4 x^{2} + 4 x + 1} = \frac{- 2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.2.2

Cancel the common factor of $- 4 x^{2} + 4 x + 1$ .

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Step 11.3.2.4.2.2.1

Cancel the common factor.

$\frac{v (- 4 x^{2} + 4 x + 1)}{- 4 x^{2} + 4 x + 1} = \frac{- 2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.2.2.2

Divide $v$ by $1$ .

$v = \frac{- 2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.3

Simplify the right side.

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Step 11.3.2.4.3.1

Move the negative in front of the fraction.

$v = - \frac{2 d}{11 (- 4 x^{2} + 4 x + 1)}$

Step 11.3.2.4.3.2

Factor $- 1$ out of $- 4 x^{2}$ .

$v = - \frac{2 d}{11 (- (4 x^{2}) + 4 x + 1)}$

Step 11.3.2.4.3.3

Factor $- 1$ out of $4 x$ .

$v = - \frac{2 d}{11 (- (4 x^{2}) - (- 4 x) + 1)}$

Step 11.3.2.4.3.4

Factor $- 1$ out of $- (4 x^{2}) - (- 4 x)$ .

$v = - \frac{2 d}{11 (- (4 x^{2} - 4 x) + 1)}$

Step 11.3.2.4.3.5

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

$v = - \frac{2 d}{11 (- (4 x^{2} - 4 x) - 1 (- 1))}$

Step 11.3.2.4.3.6

Factor $- 1$ out of $- (4 x^{2} - 4 x) - 1 (- 1)$ .

$v = - \frac{2 d}{11 (- (4 x^{2} - 4 x - 1))}$

Step 11.3.2.4.3.7

Simplify the expression.

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Step 11.3.2.4.3.7.1

Rewrite $- (4 x^{2} - 4 x - 1)$ as $- 1 (4 x^{2} - 4 x - 1)$ .

$v = - \frac{2 d}{11 (- 1 (4 x^{2} - 4 x - 1))}$

Step 11.3.2.4.3.7.2

Move the negative in front of the fraction.

$v = - - \frac{2 d}{11 (4 x^{2} - 4 x - 1)}$

Step 11.3.2.4.3.7.3

Multiply $- 1$ by $- 1$ .

$v = 1 \frac{2 d}{11 (4 x^{2} - 4 x - 1)}$

Step 11.3.2.4.3.7.4

Multiply $\frac{2 d}{11 (4 x^{2} - 4 x - 1)}$ by $1$ .

$v = \frac{2 d}{11 (4 x^{2} - 4 x - 1)}$

Step 11.4

The final solution is all the values that make $11 v (11 v - 44 v x^{2} + 44 v x + 2 d) \geq 0$ true.

$v = 0$

$v = \frac{2 d}{11 (4 x^{2} - 4 x - 1)}$

$v = 0$

$v = \frac{2 d}{11 (4 x^{2} - 4 x - 1)}$

Step 12

Set the denominator in $\frac{11 v + \sqrt{11 v (11 v - 44 v x^{2} + 44 v x + 2 d)}}{22 v}$ equal to $0$ to find where the expression is undefined.

$22 v = 0$

Step 13

Divide each term in $22 v = 0$ by $22$ and simplify.

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Step 13.1

Divide each term in $22 v = 0$ by $22$ .

$\frac{22 v}{22} = \frac{0}{22}$

Step 13.2

Simplify the left side.

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Step 13.2.1

Cancel the common factor of $22$ .

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Step 13.2.1.1

Cancel the common factor.

$\frac{22 v}{22} = \frac{0}{22}$

Step 13.2.1.2

Divide $v$ by $1$ .

$v = \frac{0}{22}$

Step 13.3

Simplify the right side.

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Step 13.3.1

Divide $0$ by $22$ .

$v = 0$

Step 14

The domain is all values of $v$ that make the expression defined.

$(N o (Min i m u m), N o (Max i m u m)]$

Set-Builder Notation:

${v | N o (Min i m u m) < v \leq N o (Max i m u m)}$