Questions: Quiz Solve the following equation for x. e^(x^2-4x)=e^12 x= (Type an integer or a decimal. Use a comma to

Solution Steps

To solve the equation \( e^{x^2 - 4x} = e^{12} \), we can take the natural logarithm of both sides to simplify the equation. This will allow us to solve for \( x \) by setting the exponents equal to each other and solving the resulting quadratic equation.

1. Take the natural logarithm of both sides of the equation.
2. Set the exponents equal to each other.
3. Solve the resulting quadratic equation for \( x \).

Given the equation: \[ e^{x^2 - 4x} = e^{12} \] we take the natural logarithm of both sides: \[ \ln(e^{x^2 - 4x}) = \ln(e^{12}) \]

Using the property of logarithms \(\ln(e^y) = y\), we simplify: \[ x^2 - 4x = 12 \]

Rearrange the equation to standard quadratic form: \[ x^2 - 4x - 12 = 0 \] We solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -12\): \[ x = \frac{4 \pm \sqrt{16 + 48}}{2} \] \[ x = \frac{4 \pm \sqrt{64}}{2} \] \[ x = \frac{4 \pm 8}{2} \] This gives us two solutions: \[ x = \frac{4 + 8}{2} = 6 \] \[ x = \frac{4 - 8}{2} = -2 \]

Final Answer