Basic Math Examples

$y + 3 = \frac{1}{8} \cdot {(y - 4)}^{2}$

Step 1

Simplify $\frac{1}{8} \cdot {(y - 4)}^{2}$ .

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Step 1.1

Rewrite.

$y + 3 = 0 + 0 + \frac{1}{8} \cdot {(y - 4)}^{2}$

Step 1.2

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

$y + 3 = \frac{1}{8} \cdot ((y - 4) (y - 4))$

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

$y + 3 = \frac{1}{8} \cdot (y (y - 4) - 4 (y - 4))$

Step 1.3.2

Apply the distributive property.

$y + 3 = \frac{1}{8} \cdot (y \cdot y + y \cdot - 4 - 4 (y - 4))$

Step 1.3.3

Apply the distributive property.

$y + 3 = \frac{1}{8} \cdot (y \cdot y + y \cdot - 4 - 4 y - 4 \cdot - 4)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $y$ by $y$ .

$y + 3 = \frac{1}{8} \cdot (y^{2} + y \cdot - 4 - 4 y - 4 \cdot - 4)$

Step 1.4.1.2

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

$y + 3 = \frac{1}{8} \cdot (y^{2} - 4 \cdot y - 4 y - 4 \cdot - 4)$

Step 1.4.1.3

Multiply $- 4$ by $- 4$ .

$y + 3 = \frac{1}{8} \cdot (y^{2} - 4 y - 4 y + 16)$

Step 1.4.2

Subtract $4 y$ from $- 4 y$ .

$y + 3 = \frac{1}{8} \cdot (y^{2} - 8 y + 16)$

Step 1.5

Apply the distributive property.

$y + 3 = \frac{1}{8} y^{2} + \frac{1}{8} (- 8 y) + \frac{1}{8} \cdot 16$

Step 1.6

Simplify.

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Step 1.6.1

Combine $\frac{1}{8}$ and $y^{2}$ .

$y + 3 = \frac{y^{2}}{8} + \frac{1}{8} (- 8 y) + \frac{1}{8} \cdot 16$

Step 1.6.2

Cancel the common factor of $8$ .

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Step 1.6.2.1

Factor $8$ out of $- 8 y$ .

$y + 3 = \frac{y^{2}}{8} + \frac{1}{8} (8 (- y)) + \frac{1}{8} \cdot 16$

Step 1.6.2.2

Cancel the common factor.

$y + 3 = \frac{y^{2}}{8} + \frac{1}{8} (8 (- y)) + \frac{1}{8} \cdot 16$

Step 1.6.2.3

Rewrite the expression.

$y + 3 = \frac{y^{2}}{8} - y + \frac{1}{8} \cdot 16$

Step 1.6.3

Cancel the common factor of $8$ .

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Step 1.6.3.1

Factor $8$ out of $16$ .

$y + 3 = \frac{y^{2}}{8} - y + \frac{1}{8} \cdot (8 (2))$

Step 1.6.3.2

Cancel the common factor.

$y + 3 = \frac{y^{2}}{8} - y + \frac{1}{8} \cdot (8 \cdot 2)$

Step 1.6.3.3

Rewrite the expression.

$y + 3 = \frac{y^{2}}{8} - y + 2$

Step 2

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\frac{y^{2}}{8} - y + 2 = y + 3$

Step 3

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

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Step 3.1

Subtract $y$ from both sides of the equation.

$\frac{y^{2}}{8} - y + 2 - y = 3$

Step 3.2

Subtract $y$ from $- y$ .

$\frac{y^{2}}{8} - 2 y + 2 = 3$

Step 4

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

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Step 4.1

Subtract $3$ from both sides of the equation.

$\frac{y^{2}}{8} - 2 y + 2 - 3 = 0$

Step 4.2

Subtract $3$ from $2$ .

$\frac{y^{2}}{8} - 2 y - 1 = 0$

Step 5

Multiply through by the least common denominator $8$ , then simplify.

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Step 5.1

Apply the distributive property.

$8 (\frac{y^{2}}{8}) + 8 (- 2 y) + 8 \cdot - 1 = 0$

Step 5.2

Simplify.

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Step 5.2.1

Cancel the common factor of $8$ .

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Step 5.2.1.1

Cancel the common factor.

$8 (\frac{y^{2}}{8}) + 8 (- 2 y) + 8 \cdot - 1 = 0$

Step 5.2.1.2

Rewrite the expression.

$y^{2} + 8 (- 2 y) + 8 \cdot - 1 = 0$

Step 5.2.2

Multiply $- 2$ by $8$ .

$y^{2} - 16 y + 8 \cdot - 1 = 0$

Step 5.2.3

Multiply $8$ by $- 1$ .

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

Step 6

Use the quadratic formula to find the solutions.

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

Step 7

Substitute the values $a = 1$ , $b = - 16$ , and $c = - 8$ into the quadratic formula and solve for $y$ .

$\frac{16 \pm \sqrt{{(- 16)}^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 8.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 8.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot - 8}}{2 \cdot 1}$

Step 8.1.2.2

Multiply $- 4$ by $- 8$ .

$y = \frac{16 \pm \sqrt{256 + 32}}{2 \cdot 1}$

Step 8.1.3

Add $256$ and $32$ .

$y = \frac{16 \pm \sqrt{288}}{2 \cdot 1}$

Step 8.1.4

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 8.1.4.1

Factor $144$ out of $288$ .

$y = \frac{16 \pm \sqrt{144 (2)}}{2 \cdot 1}$

Step 8.1.4.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{16 \pm \sqrt{12^{2} \cdot 2}}{2 \cdot 1}$

Step 8.1.5

Pull terms out from under the radical.

$y = \frac{16 \pm 12 \sqrt{2}}{2 \cdot 1}$

Step 8.2

Multiply $2$ by $1$ .

$y = \frac{16 \pm 12 \sqrt{2}}{2}$

Step 8.3

Simplify $\frac{16 \pm 12 \sqrt{2}}{2}$ .

$y = 8 \pm 6 \sqrt{2}$

Step 9

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

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot - 8}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 8$ .

$y = \frac{16 \pm \sqrt{256 + 32}}{2 \cdot 1}$

Step 9.1.3

Add $256$ and $32$ .

$y = \frac{16 \pm \sqrt{288}}{2 \cdot 1}$

Step 9.1.4

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 9.1.4.1

Factor $144$ out of $288$ .

$y = \frac{16 \pm \sqrt{144 (2)}}{2 \cdot 1}$

Step 9.1.4.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{16 \pm \sqrt{12^{2} \cdot 2}}{2 \cdot 1}$

Step 9.1.5

Pull terms out from under the radical.

$y = \frac{16 \pm 12 \sqrt{2}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$y = \frac{16 \pm 12 \sqrt{2}}{2}$

Step 9.3

Simplify $\frac{16 \pm 12 \sqrt{2}}{2}$ .

$y = 8 \pm 6 \sqrt{2}$

Step 9.4

Change the $\pm$ to $+$ .

$y = 8 + 6 \sqrt{2}$

Step 10

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

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $- 16$ to the power of $2$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$y = \frac{16 \pm \sqrt{256 - 4 \cdot - 8}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $- 8$ .

$y = \frac{16 \pm \sqrt{256 + 32}}{2 \cdot 1}$

Step 10.1.3

Add $256$ and $32$ .

$y = \frac{16 \pm \sqrt{288}}{2 \cdot 1}$

Step 10.1.4

Rewrite $288$ as $12^{2} \cdot 2$ .

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Step 10.1.4.1

Factor $144$ out of $288$ .

$y = \frac{16 \pm \sqrt{144 (2)}}{2 \cdot 1}$

Step 10.1.4.2

Rewrite $144$ as $12^{2}$ .

$y = \frac{16 \pm \sqrt{12^{2} \cdot 2}}{2 \cdot 1}$

Step 10.1.5

Pull terms out from under the radical.

$y = \frac{16 \pm 12 \sqrt{2}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$y = \frac{16 \pm 12 \sqrt{2}}{2}$

Step 10.3

Simplify $\frac{16 \pm 12 \sqrt{2}}{2}$ .

$y = 8 \pm 6 \sqrt{2}$

Step 10.4

Change the $\pm$ to $-$ .

$y = 8 - 6 \sqrt{2}$

Step 11

The final answer is the combination of both solutions.

$y = 8 + 6 \sqrt{2}, 8 - 6 \sqrt{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$y = 8 + 6 \sqrt{2}, 8 - 6 \sqrt{2}$

Decimal Form:

$y = 16.48528137 \dots, - 0.48528137 \dots$