Calculus Examples

$2 x^{2} e^{x} = 3 x e^{x}$

Step 1

Subtract $3 x e^{x}$ from both sides of the equation.

$2 x^{2} e^{x} - 3 x e^{x} = 0$

Step 2

Factor $x e^{x}$ out of $2 x^{2} e^{x} - 3 x e^{x}$ .

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Step 2.1

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

$x e^{x} (2 x) - 3 x e^{x} = 0$

Step 2.2

Factor $x e^{x}$ out of $- 3 x e^{x}$ .

$x e^{x} (2 x) + x e^{x} (- 3) = 0$

Step 2.3

Factor $x e^{x}$ out of $x e^{x} (2 x) + x e^{x} (- 3)$ .

$x e^{x} (2 x - 3) = 0$

Step 3

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

$x = 0$

$e^{x} = 0$

$2 x - 3 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

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

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Step 5.1

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

$e^{x} = 0$

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

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

Undefined

Step 5.2.3

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

No solution

Step 6

Set $2 x - 3$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Step 6.2

Solve $2 x - 3 = 0$ for $x$ .

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Step 6.2.1

Add $3$ to both sides of the equation.

$2 x = 3$

Step 6.2.2

Divide each term in $2 x = 3$ by $2$ and simplify.

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Step 6.2.2.1

Divide each term in $2 x = 3$ by $2$ .

$\frac{2 x}{2} = \frac{3}{2}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{3}{2}$

Step 6.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{2}$

Step 7

The final solution is all the values that make $x e^{x} (2 x - 3) = 0$ true.

$x = 0, \frac{3}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{3}{2}$

Decimal Form:

$x = 0, 1.5$

Mixed Number Form:

$x = 0, 1 \frac{1}{2}$