Evaluate 4( plus or minus 1/2)^3-11( plus or minus 1/2)^2-6( plus or minus 1/2)+9

$4 {(\pm \frac{1}{2})}^{3} - 11 {(\pm \frac{1}{2})}^{2} - 6 (\pm \frac{1}{2}) + 9$

Simplify each term.

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Remove the plus–minus sign on $\pm \frac{1}{2}$ because it is raised to an even power.

$4 {(\pm \frac{1}{2})}^{3} - 11 {(\frac{1}{2})}^{2} - 6 (\pm \frac{1}{2}) + 9$

Apply the product rule to $\frac{1}{2}$ .

$4 {(\pm \frac{1}{2})}^{3} - 11 \frac{1^{2}}{2^{2}} - 6 (\pm \frac{1}{2}) + 9$

One to any power is one.

$4 {(\pm \frac{1}{2})}^{3} - 11 \frac{1}{2^{2}} - 6 (\pm \frac{1}{2}) + 9$

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

$4 {(\pm \frac{1}{2})}^{3} - 11 (\frac{1}{4}) - 6 (\pm \frac{1}{2}) + 9$

Combine $- 11$ and $\frac{1}{4}$ .

$4 {(\pm \frac{1}{2})}^{3} + \frac{- 11}{4} - 6 (\pm \frac{1}{2}) + 9$

Move the negative in front of the fraction.

$4 {(\pm \frac{1}{2})}^{3} - \frac{11}{4} - 6 (\pm \frac{1}{2}) + 9$

To write $9$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$4 {(\pm \frac{1}{2})}^{3} - \frac{11}{4} + 9 \cdot \frac{4}{4} - 6 (\pm \frac{1}{2})$

Combine $9$ and $\frac{4}{4}$ .

$4 {(\pm \frac{1}{2})}^{3} - \frac{11}{4} + \frac{9 \cdot 4}{4} - 6 (\pm \frac{1}{2})$

Combine the numerators over the common denominator.

$4 {(\pm \frac{1}{2})}^{3} + \frac{- 11 + 9 \cdot 4}{4} - 6 (\pm \frac{1}{2})$

Simplify the numerator.

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Multiply $9$ by $4$ .

$4 {(\pm \frac{1}{2})}^{3} + \frac{- 11 + 36}{4} - 6 (\pm \frac{1}{2})$

Add $- 11$ and $36$ .

$4 {(\pm \frac{1}{2})}^{3} + \frac{25}{4} - 6 (\pm \frac{1}{2})$

Use the positive value of the $\pm$ to find the first solution.

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Simplify each term.

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Apply the product rule to $\frac{1}{2}$ .

$4 \frac{1^{3}}{2^{3}} + \frac{25}{4} - 6 (\frac{1}{2})$

One to any power is one.

$4 \frac{1}{2^{3}} + \frac{25}{4} - 6 (\frac{1}{2})$

Raise $2$ to the power of $3$ .

$4 (\frac{1}{8}) + \frac{25}{4} - 6 (\frac{1}{2})$

Cancel the common factor of $4$ .

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Factor $4$ out of $8$ .

$4 \frac{1}{4 (2)} + \frac{25}{4} - 6 (\frac{1}{2})$

Cancel the common factor.

$4 \frac{1}{4 \cdot 2} + \frac{25}{4} - 6 (\frac{1}{2})$

Rewrite the expression.

$\frac{1}{2} + \frac{25}{4} - 6 (\frac{1}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $- 6$ .

$\frac{1}{2} + \frac{25}{4} + 2 (- 3) \frac{1}{2}$

Cancel the common factor.

$\frac{1}{2} + \frac{25}{4} + 2 \cdot - 3 \frac{1}{2}$

Rewrite the expression.

$\frac{1}{2} + \frac{25}{4} - 3$

Find the common denominator.

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Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{1}{2} \cdot \frac{2}{2} + \frac{25}{4} - 3$

Combine.

$\frac{1 \cdot 2}{2 \cdot 2} + \frac{25}{4} - 3$

Write $- 3$ as a fraction with denominator $1$ .

$\frac{1 \cdot 2}{2 \cdot 2} + \frac{25}{4} + \frac{- 3}{1}$

Multiply $\frac{- 3}{1}$ by $\frac{4}{4}$ .

$\frac{1 \cdot 2}{2 \cdot 2} + \frac{25}{4} + \frac{- 3}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 3}{1}$ and $\frac{4}{4}$ .

$\frac{1 \cdot 2}{2 \cdot 2} + \frac{25}{4} + \frac{- 3 \cdot 4}{4}$

Multiply $2$ by $2$ .

$\frac{1 \cdot 2}{4} + \frac{25}{4} + \frac{- 3 \cdot 4}{4}$

Combine fractions.

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Combine fractions with similar denominators.

$\frac{1 \cdot 2 + 25 - 3 \cdot 4}{4}$

Multiply.

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Multiply $2$ by $1$ .

$\frac{2 + 25 - 3 \cdot 4}{4}$

Multiply $- 3$ by $4$ .

$\frac{2 + 25 - 12}{4}$

Simplify the numerator.

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Add $2$ and $25$ .

$\frac{27 - 12}{4}$

Subtract $12$ from $27$ .

$\frac{15}{4}$

Use the negative value of the $\pm$ to find the second solution.

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Simplify each term.

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Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Apply the product rule to $- \frac{1}{2}$ .

$4 ({(- 1)}^{3} {(\frac{1}{2})}^{3}) + \frac{25}{4} - 6 (- \frac{1}{2})$

Apply the product rule to $\frac{1}{2}$ .

$4 ({(- 1)}^{3} \frac{1^{3}}{2^{3}}) + \frac{25}{4} - 6 (- \frac{1}{2})$

Raise $- 1$ to the power of $3$ .

$4 (- \frac{1^{3}}{2^{3}}) + \frac{25}{4} - 6 (- \frac{1}{2})$

One to any power is one.

$4 (- \frac{1}{2^{3}}) + \frac{25}{4} - 6 (- \frac{1}{2})$

Raise $2$ to the power of $3$ .

$4 (- \frac{1}{8}) + \frac{25}{4} - 6 (- \frac{1}{2})$

Cancel the common factor of $4$ .

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Move the leading negative in $- \frac{1}{8}$ into the numerator.

$4 (\frac{- 1}{8}) + \frac{25}{4} - 6 (- \frac{1}{2})$

Factor $4$ out of $8$ .

$4 \frac{- 1}{4 (2)} + \frac{25}{4} - 6 (- \frac{1}{2})$

Cancel the common factor.

$4 \frac{- 1}{4 \cdot 2} + \frac{25}{4} - 6 (- \frac{1}{2})$

Rewrite the expression.

$\frac{- 1}{2} + \frac{25}{4} - 6 (- \frac{1}{2})$

Move the negative in front of the fraction.

$- \frac{1}{2} + \frac{25}{4} - 6 (- \frac{1}{2})$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$- \frac{1}{2} + \frac{25}{4} - 6 (\frac{- 1}{2})$

Factor $2$ out of $- 6$ .

$- \frac{1}{2} + \frac{25}{4} + 2 (- 3) \frac{- 1}{2}$

Cancel the common factor.

$- \frac{1}{2} + \frac{25}{4} + 2 \cdot - 3 \frac{- 1}{2}$

Rewrite the expression.

$- \frac{1}{2} + \frac{25}{4} - 3 \cdot - 1$

Multiply $- 3$ by $- 1$ .

$- \frac{1}{2} + \frac{25}{4} + 3$

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{2} \cdot \frac{2}{2} + \frac{25}{4} + 3$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1}{2}$ and $\frac{2}{2}$ .

$- \frac{2}{2 \cdot 2} + \frac{25}{4} + 3$

Multiply $2$ by $2$ .

$- \frac{2}{4} + \frac{25}{4} + 3$

Combine the numerators over the common denominator.

$\frac{- 2 + 25}{4} + 3$

Add $- 2$ and $25$ .

$\frac{23}{4} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{23}{4} + 3 \cdot \frac{4}{4}$

Combine $3$ and $\frac{4}{4}$ .

$\frac{23}{4} + \frac{3 \cdot 4}{4}$

Combine the numerators over the common denominator.

$\frac{23 + 3 \cdot 4}{4}$

Simplify the numerator.

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Multiply $3$ by $4$ .

$\frac{23 + 12}{4}$

Add $23$ and $12$ .

$\frac{35}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

$\frac{15}{4}, \frac{35}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{15}{4}, \frac{35}{4}$

Decimal Form:

$3.75, 8.75$

Mixed Number Form:

$3 \frac{3}{4}, 8 \frac{3}{4}$

Evaluate 4( plus or minus 1/2)^3-11( plus or minus 1/2)^2-6( plus or minus 1/2)+9

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