The quadratic equation whose roots are the A.M. and H.M. between the roots of the equation, $2x^2- 3x + 5 = 0$ is
- $4x^2 - 25x + 10 = 0$
- $12x^2 - 49x + 30 = 0$
- $14x^2 - 12x + 35 = 0$
- $2x^2 + 3x + 5 = 0$
For 2x^2 - 3x + 5 = 0, sum of roots alpha+beta = 3/2 and product alpha*beta = 5/2. AM = (alpha+beta)/2 = 3/4. HM = 2*alpha*beta/(alpha+beta) = 5/(3/2) = 10/3. The equation is x^2 - (AM+HM)x + (AM*HM) = 0, which is x^2 - (3/4 + 10/3)x + (3/4 * 10/3) = 0, leading to 12x^2 - 49x + 30 = 0.
For the given equation 2x squared minus 3x plus 5 equals 0, the sum of the roots is three halves and the product of the roots is five halves. The arithmetic mean between the roots is their sum divided by 2, which is three fourths, and the harmonic mean is 2 times the product divided by the sum, yielding 10 divided by 3. A new quadratic equation with these values as roots has a sum of 49 divided by 12 and a product of 5 divided by 2. Multiplying the standard form x squared minus the sum times x plus the product by 12 gives the equation 12x squared minus 49x plus 30 equals 0.