Question

Solve each equation. $$ e^{x^{2}}=e^{3 x} \cdot \frac{1}{e^{2}} $$

Short answer

The solutions are \( x = 1 \) and \( x = 2 \).

Step-by-step solution

Simplify the right-hand side

Rewrite the right-hand side of the equation: $$ e^{3x} \times \frac{1}{e^2} = e^{3x} \times e^{-2} $$ This can be simplified using the property of exponents, which states that when multiplying with the same base, you can add the exponents: $$ e^{3x} \times e^{-2} = e^{3x - 2} $$

Set the exponents equal

Since the bases on both sides of the equation are the same, set the exponents equal to each other: $$ x^2 = 3x - 2 $$

Rearrange and form a quadratic equation

Move all the terms to one side of the equation so it is set to zero: $$ x^2 - 3x + 2 = 0 $$

Factor the quadratic equation

Now, factorize the quadratic equation: $$ (x - 1)(x - 2) = 0 $$

Solve for x

Set each factor to zero and solve for x: $$ x - 1 = 0 \rightarrow x = 1 $$ $$ x - 2 = 0 \rightarrow x = 2 $$

Key concepts

Exponents

Exponents are a fundamental concept in algebra and mathematics in general. They indicate how many times a number, known as the base, is multiplied by itself. For example, in the expression \(e^{3x}\), 'e' is the base and '3x' is the exponent.
There are several key rules for working with exponents:

• Product Rule: \(a^m \times a^n = a^{m+n}\)
• Quotient Rule: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{mn}\)

In the given exercise, we used these rules to simplify the expressions like \(e^{3x} \times e^{-2} = e^{3x - 2}\). This simplification is crucial for setting up the equation so we can solve it effectively.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). In these equations, the highest power of the variable (usually x) is 2.
They often appear in various areas of math, such as when solving for the roots of an equation. These roots are the values of x that satisfy the equation.
In this problem, after simplifying the exponents, we ended up with a quadratic equation: \(x^2 = 3x - 2\). By rearranging the terms, we obtained: \(x^2 - 3x + 2 = 0\). This is the standard form of a quadratic equation, setting us up to solve it by factoring or using the quadratic formula.

Factoring Equations

Factoring is a method used to solve polynomial equations by breaking them down into simpler terms (factors) that, when multiplied together, give the original polynomial.
When dealing with quadratic equations, factoring can be an efficient method if the quadratic is factorable.
For our quadratic equation \(x^2 - 3x + 2 = 0\), we look for two numbers that multiply to the constant term (2) and add up to the coefficient of the middle term (-3). These numbers turn out to be -1 and -2.
We rewrite the equation as \((x - 1)(x - 2) = 0\). Once factored, the solutions are found by setting each factor equal to zero: \(x - 1 = 0\) giving us \(x = 1\) and \(x - 2 = 0\) giving us \(x = 2\).

Simplifying Exponents

Simplifying exponents involves reducing expressions with exponents to their simplest form. This often makes solving equations easier and more straightforward.
In our exercise, we simplified the right-hand side of the equation by observing that \(\frac{1}{e^2}\) is the same as \(e^{-2}\). This used the property of negative exponents: \(a^{-n} = \frac{1}{a^n}\).
By rewriting the expression, we combined the exponents when the bases were the same: \(e^{3x} \times e^{-2} = e^{3x - 2}\).
This step significantly streamlined our approach, allowing us to compare the exponents directly and transform the original equation into a manageable quadratic form.