Algebra Examples

$3 x^{\frac{3}{2}} - 9 x^{\frac{1}{2}} + 6 x^{\frac{- 1}{2}}$

Step 1

Factor $3 x^{\frac{- 1}{2}}$ out of $3 x^{\frac{3}{2}} - 9 x^{\frac{1}{2}} + 6 x^{\frac{- 1}{2}}$ .

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Step 1.1

Factor $3 x^{\frac{- 1}{2}}$ out of $3 x^{\frac{3}{2}}$ .

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

Step 1.2

Factor $3 x^{\frac{- 1}{2}}$ out of $- 9 x^{\frac{1}{2}}$ .

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

Step 1.3

Factor $3 x^{\frac{- 1}{2}}$ out of $6 x^{\frac{- 1}{2}}$ .

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

Step 1.4

Factor $3 x^{\frac{- 1}{2}}$ out of $3 x^{\frac{- 1}{2}} (x^{\frac{4}{2}}) + 3 x^{\frac{- 1}{2}} (- 3 x^{\frac{2}{2}})$ .

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

Step 1.5

Factor $3 x^{\frac{- 1}{2}}$ out of $3 x^{\frac{- 1}{2}} (x^{\frac{4}{2}} - 3 x^{\frac{2}{2}}) + 3 x^{\frac{- 1}{2}} (2)$ .

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

Step 2

Factor $x^{\frac{4}{2}} - 3 x^{\frac{2}{2}} + 2$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $2$ and whose sum is $- 3$ .

$- 2, - 1$

Step 2.2

Write the factored form using these integers.

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

Step 3

Divide $2$ by $2$ .

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

Step 4

Simplify.

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

Step 5

Rewrite $x^{\frac{2}{2}}$ as ${(x^{\frac{1}{2}})}^{2}$ .

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

Step 6

Rewrite $1$ as $1^{2}$ .

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

Step 7

Factor.

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Step 7.1

Factor.

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Step 7.1.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{\frac{1}{2}}$ and $b = 1$ .

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

Step 7.1.2

Remove unnecessary parentheses.

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

Step 7.2

Remove unnecessary parentheses.

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