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Quadratic Equation and Inequalities question

2021 · 27 Jul · Shift 2 · Q40
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  5. /2021 · 27 Jul · Shift 2 · Q40

Quadratic Equation and Inequalities question

2021 · 27 Jul · Shift 2 · Q40

JEE MainMathematicsQuadratic Equation and InequalitiesNumerical+4 / −1
The number of real roots of the equation e4x −-− e3x −-− 4e2x −-− ex + 1 = 0 is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 2

  1. Interpret the equation

    The given equation is e4x−e3x−4e2x−ex+1=0.e^{4x} - e^{3x} - 4e^{2x} - e^x + 1 = 0.e4x−e3x−4e2x−ex+1=0.

  2. Substitute to reduce it to a polynomial

    Let t=ex.t = e^x.t=ex. Since ex>0e^x > 0ex>0 for all real xxx, we must have t>0.t > 0.t>0.

    Then the equation becomes t4−t3−4t2−t+1=0.t^4 - t^3 - 4t^2 - t + 1 = 0.t4−t3−4t2−t+1=0.

  3. Factor the quartic

    We try factoring it into two quadratics: t4−t3−4t2−t+1=(t2+at−1)(t2+bt−1).t^4 - t^3 - 4t^2 - t + 1 = (t^2 + at - 1)(t^2 + bt - 1).t4−t3−4t2−t+1=(t2+at−1)(t2+bt−1).

    Expanding,

    = t^4 + (a+b)t^3 + (ab-2)t^2 - (a+b)t + 1.$$ Comparing coefficients with $$t^4 - t^3 - 4t^2 - t + 1,$$ we get $$a+b = -1, ab - 2 = -4 \,\Rightarrow\, ab = -2.$$ Numbers satisfying these are $a=1$, $b=-2$ (or vice versa). Hence $$t^4 - t^3 - 4t^2 - t + 1 = (t^2+t-1)(t^2-2t-1).$$

  4. Solve each quadratic

    From t2+t−1=0,t^2+t-1=0,t2+t−1=0, we get t=−1±52.t = \frac{-1\pm\sqrt{5}}{2}.t=2−1±5​​. Only the positive root is valid: t=5−12>0.t = \frac{\sqrt{5}-1}{2} > 0.t=25​−1​>0.

    From t2−2t−1=0,t^2-2t-1=0,t2−2t−1=0, we get t=2±4+42=1±2.t = \frac{2\pm\sqrt{4+4}}{2} = 1\pm\sqrt{2}.t=22±4+4​​=1±2​. Only the positive root is valid: t=1+2>0.t = 1+\sqrt{2} > 0.t=1+2​>0.

  5. Convert back to xxx

    Since t=ext=e^xt=ex, each positive value of ttt gives exactly one real value of xxx: x=ln⁡(5−12),x=ln⁡(1+2).x = \ln\left(\frac{\sqrt{5}-1}{2}\right), \qquad x = \ln(1+\sqrt{2}).x=ln(25​−1​),x=ln(1+2​).

    Therefore, the equation has 2 real roots.

  6. Comparison with stored answer

    My derived answer is 222, which matches the stored correct answer.

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