Solve 5e ^ 2 - x ^ 2 * e ^ (2x) + 2e ^ (2x) = 10

Answer 1

To solve the equation 5e^2 - x^2 * e^(2x) + 2e^(2x) = 10, we can rearrange the terms to get a quadratic equation in terms of e^(2x):

5e^2 - 10 = x^2 * e^(2x) - 2e^(2x)

Now, let's make a substitution u = e^(2x):

5u^2 - 10 = x^2 * u - 2u

This simplifies to:

5u^2 - x^2 * u - 2u - 10 = 0

Now we have a quadratic equation in terms of u. We can solve this equation for u and then substitute back to solve for x.

Using the quadratic formula, we have:

u = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 5, b = -x^2-2, and c = -10.

Plugging in these values, we get:

u = (x^2 + 2 ± sqrt((x^2 + 2)^2 - 4 * 5 * (-10))) / 10

Simplifying further:

u = (x^2 + 2 ± sqrt(x^4 + 4x^2 + 4 - 200)) / 10

u = (x^2 + 2 ± sqrt(x^4 + 4x^2 - 196)) / 10

Now, we can solve for x.