Algebra Examples

$9 x^{2} - 24 x + 16$

Step 1

Factor out the $9$

$9 (x^{2} - \frac{8 x}{3} + \frac{16}{9})$

Step 2

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

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $8$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

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

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Step 2.3.1

Apply the product rule to $- \frac{4}{3}$ .

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

Step 2.3.2

Apply the product rule to $\frac{4}{3}$ .

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

Step 2.4

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

$1 \frac{4^{2}}{3^{2}}$

Step 2.5

Multiply $\frac{4^{2}}{3^{2}}$ by $1$ .

$\frac{4^{2}}{3^{2}}$

Step 2.6

Raise $4$ to the power of $2$ .

$\frac{16}{3^{2}}$

Step 2.7

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

$\frac{16}{9}$

Step 3

Add $\frac{16}{9}$ to get the perfect square trinomial.

$9 (x^{2} - \frac{8 x}{3} + \frac{16}{9} + \frac{16}{9})$

Step 4

Simplify.

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Step 4.1

Combine the numerators over the common denominator.

$9 (x^{2} + \frac{- 8 x}{3} + \frac{16 + 16}{9})$

Step 4.2

Simplify the expression.

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Step 4.2.1

Add $16$ and $16$ .

$9 (x^{2} + \frac{- 8 x}{3} + \frac{32}{9})$

Step 4.2.2

Move the negative in front of the fraction.

$9 (x^{2} - \frac{8 x}{3} + \frac{32}{9})$

Step 4.3

Apply the distributive property.

$9 x^{2} + 9 (- \frac{8 x}{3}) + 9 (\frac{32}{9})$

Step 4.4

Simplify.

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Step 4.4.1

Cancel the common factor of $3$ .

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Step 4.4.1.1

Move the leading negative in $- \frac{8 x}{3}$ into the numerator.

$9 x^{2} + 9 \frac{- 8 x}{3} + 9 (\frac{32}{9})$

Step 4.4.1.2

Factor $3$ out of $9$ .

$9 x^{2} + 3 (3) \frac{- 8 x}{3} + 9 (\frac{32}{9})$

Step 4.4.1.3

Cancel the common factor.

$9 x^{2} + 3 \cdot 3 \frac{- 8 x}{3} + 9 (\frac{32}{9})$

Step 4.4.1.4

Rewrite the expression.

$9 x^{2} + 3 (- 8 x) + 9 (\frac{32}{9})$

Step 4.4.2

Multiply $- 8$ by $3$ .

$9 x^{2} - 24 x + 9 (\frac{32}{9})$

Step 4.4.3

Cancel the common factor of $9$ .

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Step 4.4.3.1

Cancel the common factor.

$9 x^{2} - 24 x + 9 (\frac{32}{9})$

Step 4.4.3.2

Rewrite the expression.

$9 x^{2} - 24 x + 32$