Precalculus Examples

$4 f - 6 f^{2}$

Step 1

Reorder $4 f$ and $- 6 f^{2}$ .

$- 6 f^{2} + 4 f$

Step 2

Factor out the $- 6$

$- 6 (- \frac{2 f}{3} + f^{2})$

Step 3

To find the $c$ value $c = {(\frac{b}{2})}^{2}$ , divide the coefficient of $f$ by $2$ and square the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

${(- (\frac{2}{3} \cdot \frac{1}{2}))}^{2}$

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

${(- (\frac{2}{3} \cdot \frac{1}{2}))}^{2}$

Step 3.2.2

Rewrite the expression.

${(- \frac{1}{3})}^{2}$

Step 3.3

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

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Step 3.3.1

Apply the product rule to $- \frac{1}{3}$ .

${(- 1)}^{2} {(\frac{1}{3})}^{2}$

Step 3.3.2

Apply the product rule to $\frac{1}{3}$ .

${(- 1)}^{2} \frac{1^{2}}{3^{2}}$

Step 3.4

Raise $- 1$ to the power of $2$ .

$1 \frac{1^{2}}{3^{2}}$

Step 3.5

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$\frac{1^{2}}{3^{2}}$

Step 3.6

One to any power is one.

$\frac{1}{3^{2}}$

Step 3.7

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

$\frac{1}{9}$

Step 4

Add $\frac{1}{9}$ to get the perfect square trinomial.

$- 6 (- \frac{2 f}{3} + f^{2} + \frac{1}{9})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$- 6 (- \frac{2 f}{3}) - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2

Simplify.

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Step 5.2.1

Cancel the common factor of $3$ .

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Step 5.2.1.1

Move the leading negative in $- \frac{2 f}{3}$ into the numerator.

$- 6 \frac{- 2 f}{3} - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2.1.2

Factor $3$ out of $- 6$ .

$3 (- 2) \frac{- 2 f}{3} - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2.1.3

Cancel the common factor.

$3 \cdot - 2 \frac{- 2 f}{3} - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2.1.4

Rewrite the expression.

$- 2 (- 2 f) - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2.2

Multiply $- 2$ by $- 2$ .

$4 f - 6 f^{2} - 6 (\frac{1}{9})$

Step 5.2.3

Cancel the common factor of $3$ .

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Step 5.2.3.1

Factor $3$ out of $- 6$ .

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

Step 5.2.3.2

Factor $3$ out of $9$ .

$4 f - 6 f^{2} + 3 \cdot - 2 \frac{1}{3 \cdot 3}$

Step 5.2.3.3

Cancel the common factor.

$4 f - 6 f^{2} + 3 \cdot - 2 \frac{1}{3 \cdot 3}$

Step 5.2.3.4

Rewrite the expression.

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

Step 5.2.4

Combine $- 2$ and $\frac{1}{3}$ .

$4 f - 6 f^{2} + \frac{- 2}{3}$

Step 5.3

Move the negative in front of the fraction.

$4 f - 6 f^{2} - \frac{2}{3}$