Multiple choice

The H.M. of the roots of the equation $\left ( 5+\sqrt{2} \right )x^{2}-\left ( 4+\sqrt{5} \right )x+\left ( 8+2\sqrt{5} \right )=0$ is

  1. 2

  2. 4

  3. 6

  4. 8
Reveal answer Fill a bubble to check yourself
B Correct answer
AI explanation

The harmonic mean of the roots of a quadratic equation is defined as 2*alpha*beta / (alpha + beta). Using Vieta's formulas for (5 + sqrt(2))x^2 - (4 + sqrt(5))x + (8 + 2*sqrt(5)) = 0, the sum of the roots alpha + beta is (4 + sqrt(5)) / (5 + sqrt(2)) and the product alpha*beta is (8 + 2*sqrt(5)) / (5 + sqrt(2)). Factoring a 2 out of the numerator of the product gives 2*(4 + sqrt(5)) / (5 + sqrt(2)). Substituting these values into the harmonic mean formula gives 2 * (2*(4 + sqrt(5)) / (5 + sqrt(2))) / ((4 + sqrt(5)) / (5 + sqrt(2))). The terms (4 + sqrt(5)) and (5 + sqrt(2)) cancel out from the numerator and denominator, leaving 4. The result is 4.

Keep practicing — related questions