Multiple choice

Find the value of $\dfrac{a}{b} + \dfrac{b}{a}$, if $a$ and $b$ are the roots of the quadratic equation $x^2+8x+4=0?$

  1. $14$
  2. $15$
  3. $24$
  4. $26$
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A Correct answer
Explanation

For a quadratic equation ax^2 + bx + c = 0, the sum of roots a+b = -b/a and product ab = c/a. Here, a+b = -8 and ab = 4. The expression a/b + b/a simplifies to (a^2 + b^2) / ab, which is ((a+b)^2 - 2ab) / ab. Substituting values: ((-8)^2 - 2*4) / 4 = (64 - 8) / 4 = 56 / 4 = 14.

AI explanation

For the quadratic equation x^2 + 8x + 4 = 0, the sum of the roots a and b is -8 and the product of the roots is 4. The expression (a/b) + (b/a) can be combined into a single fraction as (a^2 + b^2) / (a * b). Using the identity a^2 + b^2 = (a + b)^2 - 2ab, we substitute the known values to get (-8)^2 - 2(4), which equals 64 - 8 = 56. Dividing this by the product ab = 4 gives 56 / 4 = 14.

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