Pre-Algebra Examples

${(y + 4)}^{2} = 2 y^{2} + 16 + 28$

Step 1

Add $16$ and $28$ .

${(y + 4)}^{2} = 2 y^{2} + 44$

Step 2

Move all terms containing $y$ to the left side of the equation.

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Step 2.1

Subtract $2 y^{2}$ from both sides of the equation.

${(y + 4)}^{2} - 2 y^{2} = 44$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(y + 4)}^{2}$ as $(y + 4) (y + 4)$ .

$(y + 4) (y + 4) - 2 y^{2} = 44$

Step 2.2.2

Expand $(y + 4) (y + 4)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$y (y + 4) + 4 (y + 4) - 2 y^{2} = 44$

Step 2.2.2.2

Apply the distributive property.

$y \cdot y + y \cdot 4 + 4 (y + 4) - 2 y^{2} = 44$

Step 2.2.2.3

Apply the distributive property.

$y \cdot y + y \cdot 4 + 4 y + 4 \cdot 4 - 2 y^{2} = 44$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot 4 + 4 y + 4 \cdot 4 - 2 y^{2} = 44$

Step 2.2.3.1.2

Move $4$ to the left of $y$ .

$y^{2} + 4 \cdot y + 4 y + 4 \cdot 4 - 2 y^{2} = 44$

Step 2.2.3.1.3

Multiply $4$ by $4$ .

$y^{2} + 4 y + 4 y + 16 - 2 y^{2} = 44$

Step 2.2.3.2

Add $4 y$ and $4 y$ .

$y^{2} + 8 y + 16 - 2 y^{2} = 44$

Step 2.3

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

$- y^{2} + 8 y + 16 = 44$

Step 3

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

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Step 3.1

Subtract $44$ from both sides of the equation.

$- y^{2} + 8 y + 16 - 44 = 0$

Step 3.2

Subtract $44$ from $16$ .

$- y^{2} + 8 y - 28 = 0$

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 $a = - 1$ , $b = 8$ , and $c = - 28$ into the quadratic formula and solve for $y$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (- 1 \cdot - 28)}}{2 \cdot - 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 28}}{2 \cdot - 1}$

Step 6.1.2

Multiply $- 4 \cdot - 1 \cdot - 28$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 8 \pm \sqrt{64 + 4 \cdot - 28}}{2 \cdot - 1}$

Step 6.1.2.2

Multiply $4$ by $- 28$ .

$y = \frac{- 8 \pm \sqrt{64 - 112}}{2 \cdot - 1}$

Step 6.1.3

Subtract $112$ from $64$ .

$y = \frac{- 8 \pm \sqrt{- 48}}{2 \cdot - 1}$

Step 6.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot - 1}$

Step 6.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot - 1}$

Step 6.1.6

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

$y = \frac{- 8 \pm i \cdot \sqrt{48}}{2 \cdot - 1}$

Step 6.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 6.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot - 1}$

Step 6.1.7.2

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

$y = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Step 6.1.8

Pull terms out from under the radical.

$y = \frac{- 8 \pm i \cdot (4 \sqrt{3})}{2 \cdot - 1}$

Step 6.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{2 \cdot - 1}$

Step 6.2

Multiply $2$ by $- 1$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{- 2}$

Step 6.3

Simplify $\frac{- 8 \pm 4 i \sqrt{3}}{- 2}$ .

$y = 4 \pm 2 i \sqrt{3}$

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

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 28}}{2 \cdot - 1}$

Step 7.1.2

Multiply $- 4 \cdot - 1 \cdot - 28$ .

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Step 7.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 8 \pm \sqrt{64 + 4 \cdot - 28}}{2 \cdot - 1}$

Step 7.1.2.2

Multiply $4$ by $- 28$ .

$y = \frac{- 8 \pm \sqrt{64 - 112}}{2 \cdot - 1}$

Step 7.1.3

Subtract $112$ from $64$ .

$y = \frac{- 8 \pm \sqrt{- 48}}{2 \cdot - 1}$

Step 7.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot - 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot - 1}$

Step 7.1.6

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

$y = \frac{- 8 \pm i \cdot \sqrt{48}}{2 \cdot - 1}$

Step 7.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 7.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot - 1}$

Step 7.1.7.2

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

$y = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Step 7.1.8

Pull terms out from under the radical.

$y = \frac{- 8 \pm i \cdot (4 \sqrt{3})}{2 \cdot - 1}$

Step 7.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{2 \cdot - 1}$

Step 7.2

Multiply $2$ by $- 1$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{- 2}$

Step 7.3

Simplify $\frac{- 8 \pm 4 i \sqrt{3}}{- 2}$ .

$y = 4 \pm 2 i \sqrt{3}$

Step 7.4

Change the $\pm$ to $+$ .

$y = 4 + 2 i \sqrt{3}$

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

Raise $8$ to the power of $2$ .

$y = \frac{- 8 \pm \sqrt{64 - 4 \cdot - 1 \cdot - 28}}{2 \cdot - 1}$

Step 8.1.2

Multiply $- 4 \cdot - 1 \cdot - 28$ .

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Step 8.1.2.1

Multiply $- 4$ by $- 1$ .

$y = \frac{- 8 \pm \sqrt{64 + 4 \cdot - 28}}{2 \cdot - 1}$

Step 8.1.2.2

Multiply $4$ by $- 28$ .

$y = \frac{- 8 \pm \sqrt{64 - 112}}{2 \cdot - 1}$

Step 8.1.3

Subtract $112$ from $64$ .

$y = \frac{- 8 \pm \sqrt{- 48}}{2 \cdot - 1}$

Step 8.1.4

Rewrite $- 48$ as $- 1 (48)$ .

$y = \frac{- 8 \pm \sqrt{- 1 \cdot 48}}{2 \cdot - 1}$

Step 8.1.5

Rewrite $\sqrt{- 1 (48)}$ as $\sqrt{- 1} \cdot \sqrt{48}$ .

$y = \frac{- 8 \pm \sqrt{- 1} \cdot \sqrt{48}}{2 \cdot - 1}$

Step 8.1.6

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

$y = \frac{- 8 \pm i \cdot \sqrt{48}}{2 \cdot - 1}$

Step 8.1.7

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 8.1.7.1

Factor $16$ out of $48$ .

$y = \frac{- 8 \pm i \cdot \sqrt{16 (3)}}{2 \cdot - 1}$

Step 8.1.7.2

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

$y = \frac{- 8 \pm i \cdot \sqrt{4^{2} \cdot 3}}{2 \cdot - 1}$

Step 8.1.8

Pull terms out from under the radical.

$y = \frac{- 8 \pm i \cdot (4 \sqrt{3})}{2 \cdot - 1}$

Step 8.1.9

Move $4$ to the left of $i$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{2 \cdot - 1}$

Step 8.2

Multiply $2$ by $- 1$ .

$y = \frac{- 8 \pm 4 i \sqrt{3}}{- 2}$

Step 8.3

Simplify $\frac{- 8 \pm 4 i \sqrt{3}}{- 2}$ .

$y = 4 \pm 2 i \sqrt{3}$

Step 8.4

Change the $\pm$ to $-$ .

$y = 4 - 2 i \sqrt{3}$

Step 9

The final answer is the combination of both solutions.

$y = 4 + 2 i \sqrt{3}, 4 - 2 i \sqrt{3}$