Answer:
Explanation:
To solve the equation e^(2x) - e^x - 42 = 0 for x, we can make the substitution y = e^x. Then the equation becomes a quadratic equation in y:
y^2 - y - 42 = 0
We can factor the quadratic equation to get:
(y - 7)(y + 6) = 0
So y = 7 or y = -6. But y = e^x, so we have:
e^x = 7 or e^x = -6
The second equation has no real solutions, since e^x is always positive. So we have:
e^x = 7
Taking the natural logarithm of both sides, we get:
ln(e^x) = ln(7)
x = ln(7)
Rounding to four decimal places, we get:
x ≈ 1.9459
Therefore, the solution to the equation e^(2x) - e^x - 42 = 0 is x ≈ 1.9459.