Multiple choice

Let $\alpha (a) $ and $\beta (a) $ be the roots of the equation $( \sqrt[3]{1+a} -1)x^2+ (\sqrt{1+a} -1) x+ (\sqrt[6]{1+a }-1)=0$ where a> -1. Then $\lim_\limits{a\to 0^+}\alpha (a)$ and $\lim_\limits{a\to 0^+}\beta (a)$ are

  1. $-\dfrac{5}{2} $ and $1$
  2. $-\dfrac{1}{2} $ and $-1$
  3. $-\dfrac{7}{2} $ and $2$
  4. $-\dfrac{9}{2} $ and $3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Using the limit as a approaches 0, we approximate (1+a)^n - 1 as n*a. The equation becomes (a/3)x^2 + (a/2)x + (a/6) = 0. Dividing by a/6 gives 2x^2 + 3x + 1 = 0. The roots are (2x+1)(x+1) = 0, so x = -1/2 and x = -1.

AI explanation

We evaluate the limit as 'a' approaches 0 by substituting 'a' with a small parameter 'h' and using the standard approximations for roots, such as the binomial expansion, where (1 + h)^(1/n) approximates 1 + h/n. Substituting these approximations into the equation gives (h/3)x^2 + (h/2)x + (h/6) = 0. Dividing the entire equation by h/6 yields the simplified polynomial 2x^2 + 3x + 1 = 0. Factoring this quadratic equation gives (2x + 1)(x + 1) = 0, resulting in the roots -1/2 and -1.

Keep practicing — related questions