For $t > -1$, let $\alpha_t$ and $\beta_t$ be the roots of the equation $$\left((t + 2)^{\frac{1}{7}} - 1\right)x^2 + \left((t + 2)^{\frac{1}{6}} - 1\right)x + \left((t + 2)^{\frac{1}{21}} - 1\right) = 0.$$ If $\lim_{t \to -1^+} \alpha_t = a$ and $\lim_{t \to -1^+} \beta_t = b$, then $72(a + b)^2$ is equal to:

Step 1: Simplify coefficients using substitution

To simplify the limit calculation, we introduce a substitution. Let $y = t + 1$. As $t$ approaches $-1$ from the right side (denoted as $t \to -1^+$), $y$ will approach $0$ from the positive side (denoted as $y \to 0^+$). Substituting $t+2 = y+1$ into the given quadratic equation transforms it into an expression in terms of $y$.

Step 2: Apply L'Hopital's Rule for coefficients

We need to find the limits of the coefficients as $y \to 0^+$. Each coefficient is of the form $\lim_{y \to 0^+} \frac{(y+1)^k - 1}{y}$. This is an indeterminate form $\frac{0}{0}$, so we can apply L'Hopital's Rule. Alternatively, we can use the binomial approximation $(1+y)^k \approx 1 + ky$ for small $y$. Applying this, we find the limits of the coefficients $A, B, C$.

Step 3: Determine the limiting quadratic equation

As $y \to 0^+$, the original quadratic equation approaches a new quadratic equation whose coefficients are the limits we just calculated. We substitute the values of $A, B, C$ into the general quadratic form $Ax^2 + Bx + C = 0$. To clear the fractions, we multiply the entire equation by the least common multiple of the denominators, which is 42.

Step 4: Find the sum of the roots

For a quadratic equation in the standard form $ax^2 + bx + c = 0$, the sum of its roots is given by the formula $-\frac{b}{a}$. In our limiting equation, $6x^2 + 7x + 2 = 0$, we identify $a=6$ and $b=7$. Therefore, the sum of the roots, which are $a$ and $b$ in the problem statement, is $-\frac{7}{6}$.

Step 5: Calculate the final expression

The problem asks for the value of $72(a+b)^2$. We have already found that $a+b = -\frac{7}{6}$. We substitute this value into the expression and perform the calculation. Squaring $-\frac{7}{6}$ gives $\frac{49}{36}$. Then, we multiply $72$ by $\frac{49}{36}$, which simplifies to $2 \times 49 = 98$.

More from this chapter

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