Multiple choice

If a,b are the roots of the equation $x^{2}+2x+4=0,$ then $\frac{1}{a^{2}}+\frac{1}{b^{2}}=$

  1. $-\frac{1}{2}$
  2. $-\frac{1}{4}$

  3. 32

  4. 34
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For x^2 + 2x + 4 = 0, a+b = -2 and ab = 4. We need 1/a^2 + 1/b^2 = (a^2 + b^2) / (ab)^2. Since a^2 + b^2 = (a+b)^2 - 2ab = (-2)^2 - 2(4) = 4 - 8 = -4. Thus, -4 / (4^2) = -4 / 16 = -1/4.

AI explanation

From the given equation x^2 + 2x + 4 = 0, the sum of the roots a + b equals -2 and the product of the roots ab equals 4. We can rewrite the expression 1/a^2 + 1/b^2 as (a^2 + b^2) / (a^2 * b^2). Using the identity a^2 + b^2 = (a + b)^2 - 2ab, we find a^2 + b^2 = (-2)^2 - 2(4) = -4. The denominator is (ab)^2 = 16, so dividing -4 by 16 gives the final result of -1/4.

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