Algebra Examples

$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$

Step 1

Set $4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$ equal to $0$ .

$4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $2 e^{- x^{2}}$ out of $4 x^{2} e^{- x^{2}} - 2 e^{- x^{2}}$ .

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Step 2.1.1

Factor $2 e^{- x^{2}}$ out of $4 x^{2} e^{- x^{2}}$ .

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

Step 2.1.2

Factor $2 e^{- x^{2}}$ out of $- 2 e^{- x^{2}}$ .

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

Step 2.1.3

Factor $2 e^{- x^{2}}$ out of $2 e^{- x^{2}} (2 x^{2}) + 2 e^{- x^{2}} (- 1)$ .

$2 e^{- x^{2}} (2 x^{2} - 1) = 0$

Step 2.2

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

$e^{- x^{2}} = 0$

$2 x^{2} - 1 = 0$

Step 2.3

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

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Step 2.3.1

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

$e^{- x^{2}} = 0$

Step 2.3.2

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

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Step 2.3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{- x^{2}}) = \ln (0)$

Step 2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3.2.3

There is no solution for $e^{- x^{2}} = 0$

No solution

Step 2.4

Set $2 x^{2} - 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $2 x^{2} - 1$ equal to $0$ .

$2 x^{2} - 1 = 0$

Step 2.4.2

Solve $2 x^{2} - 1 = 0$ for $x$ .

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Step 2.4.2.1

Add $1$ to both sides of the equation.

$2 x^{2} = 1$

Step 2.4.2.2

Divide each term in $2 x^{2} = 1$ by $2$ and simplify.

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Step 2.4.2.2.1

Divide each term in $2 x^{2} = 1$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{1}{2}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{1}{2}$

Step 2.4.2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{2}$

Step 2.4.2.3

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

$x = \pm \sqrt{\frac{1}{2}}$

Step 2.4.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 2.4.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.4.2.4.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 2.4.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 2.4.2.4.4

Combine and simplify the denominator.

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Step 2.4.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 2.4.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 2.4.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 2.4.2.4.4.4

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

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.4.2.4.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.4.2.4.4.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.4.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.4.2.4.4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 2.4.2.4.4.6.3

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.2.4.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.4.2.4.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{2}}{2^{1}}$

Step 2.4.2.4.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{2}}{2}$

Step 2.4.2.5

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

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Step 2.4.2.5.1

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

$x = \frac{\sqrt{2}}{2}$

Step 2.4.2.5.2

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

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

Step 2.4.2.5.3

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

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

Step 2.5

The final solution is all the values that make $2 e^{- x^{2}} (2 x^{2} - 1) = 0$ true.

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

Step 3

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.70710678 \dots, - 0.70710678 \dots$

Step 4