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Chapter 3: Problem 54

Use the One-to-One Property to solve the equation for \(x\). $$e^{x^{2}+6}=e^{5 x}$$

Short Answer

Expert verified
The solutions to the equation are \(x = 2\) and \(x = 3\).

Step by step solution

01

Application of One-to-One Property

Given the equation \(e^{x^{2}+6}=e^{5 x}\), apply the One-to-One Property of exponentials. That is, if the bases are the same (which are both \(e\) here), set the exponents equal to each other. This gives \(x^{2}+6 = 5x\).
02

Rearranging the Equation

Rearrange \(x^{2}+6 = 5x\) to standard quadratic form. Subtract \(5x\) from both sides to obtain \(x^{2} - 5x + 6 = 0\).
03

Solving Quadratic Equation

Next, solve the quadratic equation \(x^{2} - 5x + 6 = 0\). This equation can be factored into \((x - 2)(x - 3) = 0\).
04

Obtaining Solutions

Setting each factor to zero lead to the roots of the quadratic equation. Hence, \(x = 2\) and \(x = 3\) are the solutions.

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