Basic Math Examples

${(z + 3 \cdot y + 1)}^{2} + z^{2} = 12 \cdot y - 4$

Step 1

Rewrite ${(z + 3 y + 1)}^{2}$ as $(z + 3 y + 1) (z + 3 y + 1)$ .

$(z + 3 y + 1) (z + 3 y + 1) + z^{2} = 12 y - 4$

Step 2

Expand $(z + 3 y + 1) (z + 3 y + 1)$ by multiplying each term in the first expression by each term in the second expression.

$z \cdot z + z (3 y) + z \cdot 1 + 3 y z + 3 y (3 y) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3

Simplify each term.

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Step 3.1

Multiply $z$ by $z$ .

$z^{2} + z (3 y) + z \cdot 1 + 3 y z + 3 y (3 y) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.2

Rewrite using the commutative property of multiplication.

$z^{2} + 3 z y + z \cdot 1 + 3 y z + 3 y (3 y) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.3

Multiply $z$ by $1$ .

$z^{2} + 3 z y + z + 3 y z + 3 y (3 y) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.4

Rewrite using the commutative property of multiplication.

$z^{2} + 3 z y + z + 3 y z + 3 \cdot (3 y \cdot y) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.5

Multiply $y$ by $y$ by adding the exponents.

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Step 3.5.1

Move $y$ .

$z^{2} + 3 z y + z + 3 y z + 3 \cdot (3 (y \cdot y)) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.5.2

Multiply $y$ by $y$ .

$z^{2} + 3 z y + z + 3 y z + 3 \cdot (3 y^{2}) + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.6

Multiply $3$ by $3$ .

$z^{2} + 3 z y + z + 3 y z + 9 y^{2} + 3 y \cdot 1 + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.7

Multiply $3$ by $1$ .

$z^{2} + 3 z y + z + 3 y z + 9 y^{2} + 3 y + 1 z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.8

Multiply $z$ by $1$ .

$z^{2} + 3 z y + z + 3 y z + 9 y^{2} + 3 y + z + 1 (3 y) + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.9

Multiply $3 y$ by $1$ .

$z^{2} + 3 z y + z + 3 y z + 9 y^{2} + 3 y + z + 3 y + 1 \cdot 1 + z^{2} = 12 y - 4$

Step 3.10

Multiply $1$ by $1$ .

$z^{2} + 3 z y + z + 3 y z + 9 y^{2} + 3 y + z + 3 y + 1 + z^{2} = 12 y - 4$

Step 4

Add $3 z y$ and $3 y z$ .

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Step 4.1

Move $z$ .

$z^{2} + 3 y z + 3 y z + z + 9 y^{2} + 3 y + z + 3 y + 1 + z^{2} = 12 y - 4$

Step 4.2

Add $3 y z$ and $3 y z$ .

$z^{2} + 6 y z + z + 9 y^{2} + 3 y + z + 3 y + 1 + z^{2} = 12 y - 4$

Step 5

Add $z$ and $z$ .

$z^{2} + 6 y z + 9 y^{2} + 3 y + 2 z + 3 y + 1 + z^{2} = 12 y - 4$

Step 6

Add $3 y$ and $3 y$ .

$z^{2} + 6 y z + 9 y^{2} + 6 y + 2 z + 1 + z^{2} = 12 y - 4$

Step 7

Add $z^{2}$ and $z^{2}$ .

$2 z^{2} + 6 y z + 9 y^{2} + 6 y + 2 z + 1 = 12 y - 4$

Step 8

Move all terms to the left side of the equation and simplify.

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Step 8.1

Move all the expressions to the left side of the equation.

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Step 8.1.1

Subtract $12 y$ from both sides of the equation.

$2 z^{2} + 6 y z + 9 y^{2} + 6 y + 2 z + 1 - 12 y = - 4$

Step 8.1.2

Add $4$ to both sides of the equation.

$2 z^{2} + 6 y z + 9 y^{2} + 6 y + 2 z + 1 - 12 y + 4 = 0$

Step 8.2

Simplify $2 z^{2} + 6 y z + 9 y^{2} + 6 y + 2 z + 1 - 12 y + 4$ .

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Step 8.2.1

Subtract $12 y$ from $6 y$ .

$2 z^{2} + 6 y z + 9 y^{2} - 6 y + 2 z + 1 + 4 = 0$

Step 8.2.2

Add $1$ and $4$ .

$2 z^{2} + 6 y z + 9 y^{2} - 6 y + 2 z + 5 = 0$

Step 9

Use the quadratic formula to find the solutions.

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

Step 10

Substitute the values $a = 2$ , $b = 6 y + 2$ , and $c = 9 y^{2} - 6 y + 5$ into the quadratic formula and solve for $z$ .

$\frac{- (6 y + 2) \pm \sqrt{{(6 y + 2)}^{2} - 4 \cdot (2 \cdot (9 y^{2} - 6 y + 5))}}{2 \cdot 2}$

Step 11

Simplify.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Apply the distributive property.

$z = \frac{- (6 y) - 1 \cdot 2 \pm \sqrt{{(6 y + 2)}^{2} - 4 \cdot 2 \cdot (9 y^{2} - 6 y + 5)}}{2 \cdot 2}$

Step 11.1.2

Multiply $6$ by $- 1$ .

$z = \frac{- 6 y - 1 \cdot 2 \pm \sqrt{{(6 y + 2)}^{2} - 4 \cdot 2 \cdot (9 y^{2} - 6 y + 5)}}{2 \cdot 2}$

Step 11.1.3

Multiply $- 1$ by $2$ .

$z = \frac{- 6 y - 2 \pm \sqrt{{(6 y + 2)}^{2} - 4 \cdot 2 \cdot (9 y^{2} - 6 y + 5)}}{2 \cdot 2}$

Step 11.1.4

Add parentheses.

$z = \frac{- 6 y - 2 \pm \sqrt{{(6 y + 2)}^{2} - 4 \cdot (2 \cdot (9 y^{2} - 6 y + 5))}}{2 \cdot 2}$

Step 11.1.5

Let $u = 2 \cdot (9 y^{2} - 6 y + 5)$ . Substitute $u$ for all occurrences of $2 \cdot (9 y^{2} - 6 y + 5)$ .

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Step 11.1.5.1

Rewrite ${(6 y + 2)}^{2}$ as $(6 y + 2) (6 y + 2)$ .

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

Step 11.1.5.2

Expand $(6 y + 2) (6 y + 2)$ using the FOIL Method.

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Step 11.1.5.2.1

Apply the distributive property.

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

Step 11.1.5.2.2

Apply the distributive property.

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

Step 11.1.5.2.3

Apply the distributive property.

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

Step 11.1.5.3

Simplify and combine like terms.

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Step 11.1.5.3.1

Simplify each term.

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Step 11.1.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 11.1.5.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 11.1.5.3.1.2.1

Move $y$ .

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

Step 11.1.5.3.1.2.2

Multiply $y$ by $y$ .

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

Step 11.1.5.3.1.3

Multiply $6$ by $6$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 6 y \cdot 2 + 2 (6 y) + 2 \cdot 2 - 4 \cdot u}}{2 \cdot 2}$

Step 11.1.5.3.1.4

Multiply $2$ by $6$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 12 y + 2 (6 y) + 2 \cdot 2 - 4 \cdot u}}{2 \cdot 2}$

Step 11.1.5.3.1.5

Multiply $6$ by $2$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 12 y + 12 y + 2 \cdot 2 - 4 \cdot u}}{2 \cdot 2}$

Step 11.1.5.3.1.6

Multiply $2$ by $2$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 12 y + 12 y + 4 - 4 \cdot u}}{2 \cdot 2}$

Step 11.1.5.3.2

Add $12 y$ and $12 y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 24 y + 4 - 4 \cdot u}}{2 \cdot 2}$

$z = \frac{- 6 y - 2 \pm \sqrt{36 y^{2} + 24 y + 4 - 4 u}}{2 \cdot 2}$

Step 11.1.6

Factor $4$ out of $36 y^{2} + 24 y + 4 - 4 u$ .

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Step 11.1.6.1

Factor $4$ out of $36 y^{2}$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2}) + 24 y + 4 - 4 u}}{2 \cdot 2}$

Step 11.1.6.2

Factor $4$ out of $24 y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2}) + 4 (6 y) + 4 - 4 u}}{2 \cdot 2}$

Step 11.1.6.3

Factor $4$ out of $4$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2}) + 4 (6 y) + 4 (1) - 4 u}}{2 \cdot 2}$

Step 11.1.6.4

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

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2}) + 4 (6 y) + 4 (1) + 4 (- u)}}{2 \cdot 2}$

Step 11.1.6.5

Factor $4$ out of $4 (9 y^{2}) + 4 (6 y)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y) + 4 (1) + 4 (- u)}}{2 \cdot 2}$

Step 11.1.6.6

Factor $4$ out of $4 (9 y^{2} + 6 y) + 4 (1)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1) + 4 (- u)}}{2 \cdot 2}$

Step 11.1.6.7

Factor $4$ out of $4 (9 y^{2} + 6 y + 1) + 4 (- u)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - u)}}{2 \cdot 2}$

Step 11.1.7

Replace all occurrences of $u$ with $2 \cdot (9 y^{2} - 6 y + 5)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (2 \cdot (9 y^{2} - 6 y + 5)))}}{2 \cdot 2}$

Step 11.1.8

Simplify.

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Step 11.1.8.1

Simplify each term.

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Step 11.1.8.1.1

Apply the distributive property.

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (2 (9 y^{2}) + 2 (- 6 y) + 2 \cdot 5))}}{2 \cdot 2}$

Step 11.1.8.1.2

Simplify.

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Step 11.1.8.1.2.1

Multiply $9$ by $2$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (18 y^{2} + 2 (- 6 y) + 2 \cdot 5))}}{2 \cdot 2}$

Step 11.1.8.1.2.2

Multiply $- 6$ by $2$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (18 y^{2} - 12 y + 2 \cdot 5))}}{2 \cdot 2}$

Step 11.1.8.1.2.3

Multiply $2$ by $5$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (18 y^{2} - 12 y + 10))}}{2 \cdot 2}$

Step 11.1.8.1.3

Apply the distributive property.

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - (18 y^{2}) - (- 12 y) - 1 \cdot 10)}}{2 \cdot 2}$

Step 11.1.8.1.4

Simplify.

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Step 11.1.8.1.4.1

Multiply $18$ by $- 1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - 18 y^{2} - (- 12 y) - 1 \cdot 10)}}{2 \cdot 2}$

Step 11.1.8.1.4.2

Multiply $- 12$ by $- 1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - 18 y^{2} + 12 y - 1 \cdot 10)}}{2 \cdot 2}$

Step 11.1.8.1.4.3

Multiply $- 1$ by $10$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 y^{2} + 6 y + 1 - 18 y^{2} + 12 y - 10)}}{2 \cdot 2}$

Step 11.1.8.2

Subtract $18 y^{2}$ from $9 y^{2}$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (- 9 y^{2} + 6 y + 1 + 12 y - 10)}}{2 \cdot 2}$

Step 11.1.8.3

Add $6 y$ and $12 y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (- 9 y^{2} + 18 y + 1 - 10)}}{2 \cdot 2}$

Step 11.1.8.4

Subtract $10$ from $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (- 9 y^{2} + 18 y - 9)}}{2 \cdot 2}$

Step 11.1.9

Factor $9$ out of $- 9 y^{2} + 18 y - 9$ .

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Step 11.1.9.1

Factor $9$ out of $- 9 y^{2}$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2}) + 18 y - 9)}}{2 \cdot 2}$

Step 11.1.9.2

Factor $9$ out of $18 y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2}) + 9 (2 y) - 9)}}{2 \cdot 2}$

Step 11.1.9.3

Factor $9$ out of $- 9$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2}) + 9 (2 y) + 9 (- 1))}}{2 \cdot 2}$

Step 11.1.9.4

Factor $9$ out of $9 (- y^{2}) + 9 (2 y)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + 2 y) + 9 (- 1))}}{2 \cdot 2}$

Step 11.1.9.5

Factor $9$ out of $9 (- y^{2} + 2 y) + 9 (- 1)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + 2 y - 1))}}{2 \cdot 2}$

Step 11.1.10

Factor by grouping.

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Step 11.1.10.1

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

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Step 11.1.10.1.1

Factor $2$ out of $2 y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + 2 (y) - 1))}}{2 \cdot 2}$

Step 11.1.10.1.2

Rewrite $2$ as $1$ plus $1$

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + (1 + 1) y - 1))}}{2 \cdot 2}$

Step 11.1.10.1.3

Apply the distributive property.

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + 1 y + 1 y - 1))}}{2 \cdot 2}$

Step 11.1.10.1.4

Multiply $y$ by $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + y + 1 y - 1))}}{2 \cdot 2}$

Step 11.1.10.1.5

Multiply $y$ by $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y^{2} + y + y - 1))}}{2 \cdot 2}$

Step 11.1.10.2

Factor out the greatest common factor from each group.

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Step 11.1.10.2.1

Group the first two terms and the last two terms.

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 ((- y^{2} + y) + y - 1))}}{2 \cdot 2}$

Step 11.1.10.2.2

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

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (y (- y + 1) - 1 (- y + 1)))}}{2 \cdot 2}$

Step 11.1.10.3

Factor the polynomial by factoring out the greatest common factor, $- y + 1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 ((- y + 1) (y - 1)))}}{2 \cdot 2}$

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- y + 1) (y - 1))}}{2 \cdot 2}$

Step 11.1.11

Combine exponents.

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Step 11.1.11.1

Factor $- 1$ out of $- y$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- (y) + 1) (y - 1))}}{2 \cdot 2}$

Step 11.1.11.2

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

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- (y) - 1 \cdot - 1) (y - 1))}}{2 \cdot 2}$

Step 11.1.11.3

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- (y - 1)) (y - 1))}}{2 \cdot 2}$

Step 11.1.11.4

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

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 (- 1 (y - 1)) (y - 1))}}{2 \cdot 2}$

Step 11.1.11.5

Raise $y - 1$ to the power of $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 \cdot (- 1 ((y - 1) (y - 1))))}}{2 \cdot 2}$

Step 11.1.11.6

Raise $y - 1$ to the power of $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 \cdot (- 1 ((y - 1) (y - 1))))}}{2 \cdot 2}$

Step 11.1.11.7

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

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 \cdot (- 1 {(y - 1)}^{1 + 1}))}}{2 \cdot 2}$

Step 11.1.11.8

Add $1$ and $1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (9 \cdot (- 1 {(y - 1)}^{2}))}}{2 \cdot 2}$

Step 11.1.11.9

Multiply $9$ by $- 1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{4 (- 9 {(y - 1)}^{2})}}{2 \cdot 2}$

$z = \frac{- 6 y - 2 \pm \sqrt{4 \cdot (- 9 {(y - 1)}^{2})}}{2 \cdot 2}$

Step 11.1.12

Multiply $4$ by $- 9$ .

$z = \frac{- 6 y - 2 \pm \sqrt{- 36 {(y - 1)}^{2}}}{2 \cdot 2}$

Step 11.1.13

Rewrite $- 36 {(y - 1)}^{2}$ as ${(6 (y - 1))}^{2} \cdot - 1$ .

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Step 11.1.13.1

Factor $36$ out of $- 36$ .

$z = \frac{- 6 y - 2 \pm \sqrt{36 (- 1) {(y - 1)}^{2}}}{2 \cdot 2}$

Step 11.1.13.2

Rewrite $36$ as $6^{2}$ .

$z = \frac{- 6 y - 2 \pm \sqrt{6^{2} \cdot (- 1 {(y - 1)}^{2})}}{2 \cdot 2}$

Step 11.1.13.3

Move $- 1$ .

$z = \frac{- 6 y - 2 \pm \sqrt{6^{2} {(y - 1)}^{2} \cdot - 1}}{2 \cdot 2}$

Step 11.1.13.4

Rewrite $6^{2} {(y - 1)}^{2}$ as ${(6 (y - 1))}^{2}$ .

$z = \frac{- 6 y - 2 \pm \sqrt{{(6 (y - 1))}^{2} \cdot - 1}}{2 \cdot 2}$

Step 11.1.14

Pull terms out from under the radical.

$z = \frac{- 6 y - 2 \pm 6 (y - 1) \sqrt{- 1}}{2 \cdot 2}$

Step 11.1.15

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

$z = \frac{- 6 y - 2 \pm 6 (y - 1) i}{2 \cdot 2}$

Step 11.1.16

Apply the distributive property.

$z = \frac{- 6 y - 2 \pm (6 y + 6 \cdot - 1) i}{2 \cdot 2}$

Step 11.1.17

Multiply $6$ by $- 1$ .

$z = \frac{- 6 y - 2 \pm (6 y - 6) i}{2 \cdot 2}$

Step 11.1.18

Apply the distributive property.

$z = \frac{- 6 y - 2 \pm (6 y i - 6 i)}{2 \cdot 2}$

Step 11.2

Multiply $2$ by $2$ .

$z = \frac{- 6 y - 2 \pm (6 y i - 6 i)}{4}$

Step 12

The final answer is the combination of both solutions.

$z = \frac{3 i y - 3 y - 1 - 3 i}{2}$

$z = - \frac{3 i y + 3 y + 1 - 3 i}{2}$