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:

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