Basic Math Examples

$(\frac{11}{2} + y) (\frac{11}{2} + y) = 8$

Step 1

Simplify $(\frac{11}{2} + y) (\frac{11}{2} + y)$ .

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Step 1.1

Expand $(\frac{11}{2} + y) (\frac{11}{2} + y)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$\frac{11}{2} (\frac{11}{2} + y) + y (\frac{11}{2} + y) = 8$

Step 1.1.2

Apply the distributive property.

$\frac{11}{2} \cdot \frac{11}{2} + \frac{11}{2} y + y (\frac{11}{2} + y) = 8$

Step 1.1.3

Apply the distributive property.

$\frac{11}{2} \cdot \frac{11}{2} + \frac{11}{2} y + y \frac{11}{2} + y \cdot y = 8$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $\frac{11}{2} \cdot \frac{11}{2}$ .

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Step 1.2.1.1.1

Multiply $\frac{11}{2}$ by $\frac{11}{2}$ .

$\frac{11 \cdot 11}{2 \cdot 2} + \frac{11}{2} y + y \frac{11}{2} + y \cdot y = 8$

Step 1.2.1.1.2

Multiply $11$ by $11$ .

$\frac{121}{2 \cdot 2} + \frac{11}{2} y + y \frac{11}{2} + y \cdot y = 8$

Step 1.2.1.1.3

Multiply $2$ by $2$ .

$\frac{121}{4} + \frac{11}{2} y + y \frac{11}{2} + y \cdot y = 8$

Step 1.2.1.2

Combine $\frac{11}{2}$ and $y$ .

$\frac{121}{4} + \frac{11 y}{2} + y \frac{11}{2} + y \cdot y = 8$

Step 1.2.1.3

Combine $y$ and $\frac{11}{2}$ .

$\frac{121}{4} + \frac{11 y}{2} + \frac{y \cdot 11}{2} + y \cdot y = 8$

Step 1.2.1.4

Move $11$ to the left of $y$ .

$\frac{121}{4} + \frac{11 y}{2} + \frac{11 \cdot y}{2} + y \cdot y = 8$

Step 1.2.1.5

Multiply $y$ by $y$ .

$\frac{121}{4} + \frac{11 y}{2} + \frac{11 y}{2} + y^{2} = 8$

Step 1.2.2

Add $\frac{11 y}{2}$ and $\frac{11 y}{2}$ .

$\frac{121}{4} + 2 \frac{11 y}{2} + y^{2} = 8$

Step 1.3

Cancel the common factor of $2$ .

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Step 1.3.1

Cancel the common factor.

$\frac{121}{4} + 2 \frac{11 y}{2} + y^{2} = 8$

Step 1.3.2

Rewrite the expression.

$\frac{121}{4} + 11 y + y^{2} = 8$

Step 2

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

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Step 2.1

Subtract $8$ from both sides of the equation.

$\frac{121}{4} + 11 y + y^{2} - 8 = 0$

Step 2.2

Simplify $\frac{121}{4} + 11 y + y^{2} - 8$ .

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Step 2.2.1

To write $- 8$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$11 y + y^{2} + \frac{121}{4} - 8 \cdot \frac{4}{4} = 0$

Step 2.2.2

Combine $- 8$ and $\frac{4}{4}$ .

$11 y + y^{2} + \frac{121}{4} + \frac{- 8 \cdot 4}{4} = 0$

Step 2.2.3

Combine the numerators over the common denominator.

$11 y + y^{2} + \frac{121 - 8 \cdot 4}{4} = 0$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $- 8$ by $4$ .

$11 y + y^{2} + \frac{121 - 32}{4} = 0$

Step 2.2.4.2

Subtract $32$ from $121$ .

$11 y + y^{2} + \frac{89}{4} = 0$

Step 3

Multiply through by the least common denominator $4$ , then simplify.

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Step 3.1

Apply the distributive property.

$4 (11 y) + 4 y^{2} + 4 (\frac{89}{4}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $11$ by $4$ .

$44 y + 4 y^{2} + 4 (\frac{89}{4}) = 0$

Step 3.2.2

Cancel the common factor of $4$ .

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Step 3.2.2.1

Cancel the common factor.

$44 y + 4 y^{2} + 4 (\frac{89}{4}) = 0$

Step 3.2.2.2

Rewrite the expression.

$44 y + 4 y^{2} + 89 = 0$

Step 3.3

Reorder $44 y$ and $4 y^{2}$ .

$4 y^{2} + 44 y + 89 = 0$

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 $a = 4$ , $b = 44$ , and $c = 89$ into the quadratic formula and solve for $y$ .

$\frac{- 44 \pm \sqrt{44^{2} - 4 \cdot (4 \cdot 89)}}{2 \cdot 4}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $44$ to the power of $2$ .

$y = \frac{- 44 \pm \sqrt{1936 - 4 \cdot 4 \cdot 89}}{2 \cdot 4}$

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot 89$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

$y = \frac{- 44 \pm \sqrt{1936 - 16 \cdot 89}}{2 \cdot 4}$

Step 6.1.2.2

Multiply $- 16$ by $89$ .

$y = \frac{- 44 \pm \sqrt{1936 - 1424}}{2 \cdot 4}$

Step 6.1.3

Subtract $1424$ from $1936$ .

$y = \frac{- 44 \pm \sqrt{512}}{2 \cdot 4}$

Step 6.1.4

Rewrite $512$ as $16^{2} \cdot 2$ .

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Step 6.1.4.1

Factor $256$ out of $512$ .

$y = \frac{- 44 \pm \sqrt{256 (2)}}{2 \cdot 4}$

Step 6.1.4.2

Rewrite $256$ as $16^{2}$ .

$y = \frac{- 44 \pm \sqrt{16^{2} \cdot 2}}{2 \cdot 4}$

Step 6.1.5

Pull terms out from under the radical.

$y = \frac{- 44 \pm 16 \sqrt{2}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$y = \frac{- 44 \pm 16 \sqrt{2}}{8}$

Step 6.3

Simplify $\frac{- 44 \pm 16 \sqrt{2}}{8}$ .

$y = \frac{- 11 \pm 4 \sqrt{2}}{2}$

Step 7

The final answer is the combination of both solutions.

$y = - \frac{11 - 4 \sqrt{2}}{2}, - \frac{11 + 4 \sqrt{2}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{11 - 4 \sqrt{2}}{2}, - \frac{11 + 4 \sqrt{2}}{2}$

Decimal Form:

$y = - 2.67157287 \dots, - 8.32842712 \dots$