Algebra Examples

$8 a^{2} - 12 a b + 9 b^{2}$

Step 1

Factor out the $8$

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2}}{8})$

Step 2

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

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Multiply $\frac{3 b}{2} \cdot \frac{1}{2}$ .

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $2$ .

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

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 $- \frac{3 b}{4}$ .

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

Step 2.3.2

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

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

Step 2.3.3

Apply the product rule to $3 b$ .

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

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

$\frac{9 b^{2}}{4^{2}}$

Step 2.7

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

$\frac{9 b^{2}}{16}$

Step 3

Add $\frac{9 b^{2}}{16}$ to get the perfect square trinomial.

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

Step 4

Simplify.

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Step 4.1

To write $\frac{9 b^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2}}{8} \cdot \frac{2}{2} + \frac{9 b^{2}}{16})$

Step 4.2

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.2.1

Multiply $\frac{9 b^{2}}{8}$ by $\frac{2}{2}$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} \cdot 2}{8 \cdot 2} + \frac{9 b^{2}}{16})$

Step 4.2.2

Multiply $8$ by $2$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} \cdot 2}{16} + \frac{9 b^{2}}{16})$

Step 4.3

Combine the numerators over the common denominator.

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} \cdot 2 + 9 b^{2}}{16})$

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor $9 b^{2}$ out of $9 b^{2} \cdot 2 + 9 b^{2}$ .

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Step 4.4.1.1

Factor $9 b^{2}$ out of $9 b^{2} \cdot 2$ .

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

Step 4.4.1.2

Factor $9 b^{2}$ out of $9 b^{2}$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} (2) + 9 b^{2} (1)}{16})$

Step 4.4.1.3

Factor $9 b^{2}$ out of $9 b^{2} (2) + 9 b^{2} (1)$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} (2 + 1)}{16})$

Step 4.4.2

Add $2$ and $1$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{9 b^{2} \cdot 3}{16})$

Step 4.4.3

Multiply $3$ by $9$ .

$8 (a^{2} - \frac{3 a b}{2} + \frac{27 b^{2}}{16})$

Step 4.5

Apply the distributive property.

$8 a^{2} + 8 (- \frac{3 a b}{2}) + 8 \frac{27 b^{2}}{16}$

Step 4.6

Simplify.

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Step 4.6.1

Cancel the common factor of $2$ .

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Step 4.6.1.1

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

$8 a^{2} + 8 \frac{- 3 a b}{2} + 8 \frac{27 b^{2}}{16}$

Step 4.6.1.2

Factor $2$ out of $8$ .

$8 a^{2} + 2 (4) \frac{- 3 a b}{2} + 8 \frac{27 b^{2}}{16}$

Step 4.6.1.3

Cancel the common factor.

$8 a^{2} + 2 \cdot 4 \frac{- 3 a b}{2} + 8 \frac{27 b^{2}}{16}$

Step 4.6.1.4

Rewrite the expression.

$8 a^{2} + 4 (- 3 a b) + 8 \frac{27 b^{2}}{16}$

Step 4.6.2

Multiply $- 3$ by $4$ .

$8 a^{2} - 12 (a b) + 8 \frac{27 b^{2}}{16}$

Step 4.6.3

Cancel the common factor of $8$ .

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Step 4.6.3.1

Factor $8$ out of $16$ .

$8 a^{2} - 12 (a b) + 8 \frac{27 b^{2}}{8 (2)}$

Step 4.6.3.2

Cancel the common factor.

$8 a^{2} - 12 (a b) + 8 \frac{27 b^{2}}{8 \cdot 2}$

Step 4.6.3.3

Rewrite the expression.

$8 a^{2} - 12 (a b) + \frac{27 b^{2}}{2}$

$8 a^{2} - 12 a b + \frac{27 b^{2}}{2}$