If the sum of all the roots of the equation \(e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0\)
is logeP, then p is equal to _____.
If the sum of all the roots of the equation \(e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0\)
is logeP, then p is equal to _____.
Correct Answer: 45
Solution and Explanation
The correct answer is 45
Let \(e^x = t\) then equation reduces to
\(t^2−11t−\frac{45}{t}+\frac{81}{2}=0\)
\(⇒ 2t^3 – 22t^2 + 81t – 45 = 0 …(i)\)
if roots of
\(e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0\)
are α, β, γ then roots of (i) will be
\(e^{α_1}e^{α_2}e^{α_3} \)
Therefore , by using product of roots
\(e^{α_1+α_2+α_3}=45\)
\(⇒ α_1 + α_2 + α_3 \)
= ln 45
⇒ p = 45
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Concepts Used:
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A polynomial that has two roots or is of degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b, and c are the real numbers.
Consider the following equation ax²+bx+c=0, where a≠0 and a, b, and c are real coefficients.
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