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Chapter 3: Problem 57

Use the One-to-One Property to solve the equation for \(x\). $$e^{x^{2}-3}=e^{2 x}$$

Short Answer

Expert verified
The solution for x in the equation is 3 or -1.

Step by step solution

01

Identify the Property to be used

Based on the equation, it can be noticed that both sides of the equation have the same base 'e'. Therefore, according to the One-to-One property, the exponents can be equated to each other.
02

Set the Exponents equal to each other and Simplify

Given that \(e^{x^{2}-3} = e^{2x}\). Now, set the exponents equal to each other, i.e. \(x^{2}-3 = 2x \). Re-arrange the equation to form a quadratic equation by subtracting \(2x\) from both sides: \(x^2 - 2x - 3 = 0\).
03

Solve the Quadratic Equation

The quadratic equation is in the form \(ax^2 + bx + c = 0\). We can now solve this equation for x using the quadratic formula \(-b 卤 \sqrt{b^2 - 4ac} / 2a\). Substituting a = 1, b = -2 and c = -3 into the formula, we get two possible solutions: \(x = 3\) or \(x = -1\).

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