Multiple choice

Directions : Select the correct alternative from the given choices. If the difference between the roots of the equation x2 - px + 2p = 0 is 3, where both the roots are positive, find the equation whose roots are 2 more than the roots of the given equation.

  1. x2 - 5x + 4 = 0

  2. x2 - 9x + 18 = 0

  3. x2 + 5x + 18 = 0

  4. x2 - 13x + 40 = 0
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For x^2 - px + 2p = 0, let roots be a and b. a+b=p, ab=2p. Given |a-b|=3, so (a-b)^2 = 9, which is (a+b)^2 - 4ab = 9. Thus p^2 - 8p - 9 = 0. Solving gives p=9 or p=-1. Since roots are positive, p=9. Roots are 3 and 6. New roots are 3+2=5 and 6+2=8. The equation is (x-5)(x-8) = x^2 - 13x + 40 = 0.

AI explanation

Using the relation between roots and coefficients, the sum of the roots equals p and the product equals 2p. The difference between the roots is 3, so squaring the difference gives the sum squared minus 4 times the product, yielding p squared minus 8p equals 9. Solving p squared minus 8p minus 9 equals 0 gives p = 9, making the original roots 6 and 3. Adding 2 to each root gives new roots of 8 and 5, so the new equation is x2 - 13x + 40 = 0.

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