Question

Solve:

$\frac{e^{x^2}}{e^x}=6$

Short answer

By solving $\frac{e^{x^2}}{e^x}=6$, the value of $x$ is $-2,3$

Step-by-step solution

Step 1. Given

An expression$\frac{e^{x^2}}{e^x}=6$

To solve the expression.

Step 2. Use the property of exponent

By using the property of exponent, $\frac{a^m}{a^n}=a^{m-n}$,

$e^{x^2-x}=6$

Step 3. Write a new equation by setting the exponents equal.

Since the bases are same, the exponents must be equal.

$$\begin{gathered} x^2-x=6 \\ {x}^2-x-6=0 \\ {x}^2-3x+2x-6=0 \\ (x-3)(x+2)=0 \\ x=-2,3 \end{gathered}$$

The value of $x$ is $-2,3$

Step 4. Check the solutions.

When $x=-2$,

$$\begin{gathered} \frac{e^{{(-2)}^2}}{e^{-2}}=e^6 \\ \frac{e^4}{e^{-2}}=e^6 \\ {e}^{4+2}=e^6 \\ {e}^6=e^6 \end{gathered}$$

When $x=3$,

$$\begin{gathered} \frac{e^{3^2}}{e^3}=e^6 \\ {e}^{9-3}=e^6 \\ {e}^6=e^6 \end{gathered}$$

Hence the solution has been checked.