Algebra Examples

$6 x^{3} {(x^{2} + 1)}^{- \frac{1}{2}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} = 0$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$6 x^{3} \frac{1}{{(x^{2} + 1)}^{\frac{1}{2}}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} = 0$

Step 1.1.2

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

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Step 1.1.2.1

Combine $6$ and $\frac{1}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

$x^{3} \frac{6}{{(x^{2} + 1)}^{\frac{1}{2}}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} = 0$

Step 1.1.2.2

Combine $x^{3}$ and $\frac{6}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{x^{3} \cdot 6}{{(x^{2} + 1)}^{\frac{1}{2}}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} = 0$

Step 1.1.3

Move $6$ to the left of $x^{3}$ .

$\frac{6 x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} = 0$

Step 1.2

To write $- 4 x {(x^{2} + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{6 x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} \cdot \frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine $- 4 x {(x^{2} + 1)}^{\frac{1}{2}}$ and $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

$\frac{6 x^{3}}{{(x^{2} + 1)}^{\frac{1}{2}}} + \frac{- 4 x {(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.3.2

Combine the numerators over the common denominator.

$\frac{6 x^{3} - 4 x {(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4

Simplify the numerator.

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Step 1.4.1

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

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Step 1.4.1.1

Factor $2 x$ out of $6 x^{3}$ .

$\frac{2 x (3 x^{2}) - 4 x {(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.1.2

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

$\frac{2 x (3 x^{2}) + 2 x (- 2 {(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.1.3

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

$\frac{2 x (3 x^{2} - 2 {(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2

Combine exponents.

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Step 1.4.2.1

Multiply ${(x^{2} + 1)}^{\frac{1}{2}}$ by ${(x^{2} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.4.2.1.1

Move ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$\frac{2 x (3 x^{2} - 2 ({(x^{2} + 1)}^{\frac{1}{2}} {(x^{2} + 1)}^{\frac{1}{2}}))}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x (3 x^{2} - 2 {(x^{2} + 1)}^{\frac{1}{2} + \frac{1}{2}})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2.1.3

Combine the numerators over the common denominator.

$\frac{2 x (3 x^{2} - 2 {(x^{2} + 1)}^{\frac{1 + 1}{2}})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2.1.4

Add $1$ and $1$ .

$\frac{2 x (3 x^{2} - 2 {(x^{2} + 1)}^{\frac{2}{2}})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2.1.5

Divide $2$ by $2$ .

$\frac{2 x (3 x^{2} - 2 {(x^{2} + 1)}^{1})}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.2.2

Simplify $- 2 {(x^{2} + 1)}^{1}$ .

$\frac{2 x (3 x^{2} - 2 (x^{2} + 1))}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.3

Simplify each term.

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Step 1.4.3.1

Apply the distributive property.

$\frac{2 x (3 x^{2} - 2 x^{2} - 2 \cdot 1)}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.3.2

Multiply $- 2$ by $1$ .

$\frac{2 x (3 x^{2} - 2 x^{2} - 2)}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 1.4.4

Subtract $2 x^{2}$ from $3 x^{2}$ .

$\frac{2 x (x^{2} - 2)}{{(x^{2} + 1)}^{\frac{1}{2}}} = 0$

Step 2

Set the numerator equal to zero.

$2 x (x^{2} - 2) = 0$

Step 3

Solve the equation for $x$ .

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Step 3.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x^{2} - 2 = 0$

Step 3.2

Set $x$ equal to $0$ .

$x = 0$

Step 3.3

Set $x^{2} - 2$ equal to $0$ and solve for $x$ .

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Step 3.3.1

Set $x^{2} - 2$ equal to $0$ .

$x^{2} - 2 = 0$

Step 3.3.2

Solve $x^{2} - 2 = 0$ for $x$ .

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Step 3.3.2.1

Add $2$ to both sides of the equation.

$x^{2} = 2$

Step 3.3.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2}$

Step 3.3.2.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.3.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{2}$

Step 3.3.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{2}$

Step 3.3.2.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{2}, - \sqrt{2}$

Step 3.4

The final solution is all the values that make $2 x (x^{2} - 2) = 0$ true.

$x = 0, \sqrt{2}, - \sqrt{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = 0, 1.41421356 \dots, - 1.41421356 \dots$