If the sum of all the roots of the equation
e²ˣ - 11eˣ - 45e⁻ˣ + 81/2 = 0 is log ₑp, then p is equal to ____________.

Solution

Maths sir · 4 steps

Put t = eˣ, but only use the cubic's real positive roots, because each t must equal some eˣ.

1. Substitute. Let t = eˣ, so t > 0. Multiply the equation by eˣ and then by 2 to clear the fraction.

   2t³ - 22t² + 81t - 90 = 0

2. Find the roots. Try t = 2: 16 - 88 + 162 - 90 = 0, so it is a root. Dividing by (t - 2) leaves 2t² - 18t + 45. Its discriminant is 324 - 360 = -36 < 0, so this quadratic has no real roots.
3. Back to x. The only real positive value is t = 2. So eˣ = 2, which gives exactly one root, x = logₑ 2.
4. So. The sum of all the roots is logₑ 2, so p = 2.

The common slip: 45. Using the product of all three t-roots, 90/2 = 45, gives p = 45. Two of those roots are complex, so they do not give real values of x.

The idea to keep. After a substitution like t = eˣ, every t-root must be real and positive to give an x-root. Check that before you reach for Vieta's product.

Answer from the NTA final answer key. Solution by PrepWiser.