Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Divide $2$ by $2$ .

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

Step 3

Simplify.

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

Step 4

Apply the distributive property.

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

Step 5

Multiply $x$ by $x$ .

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

Step 6

Multiply $x$ by $1$ .

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

Step 7

Factor.

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Step 7.1

Factor $x^{2} + x - 2$ using the AC method.

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Step 7.1.1

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 $1$ .

$- 1, 2$

Step 7.1.2

Write the factored form using these integers.

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

Step 7.2

Remove unnecessary parentheses.

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