Multiple choice

If a, b are the roots of the equation x2 – 7x + P = 0 and a – b = 3, then what is the value of P?

  1. 7

  2. 8

  3. 9

  4. 10
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D Correct answer
Explanation

Roots a, b satisfy a+b = 7 and ab = P. Given a-b = 3. (a-b)^2 = (a+b)^2 - 4ab. 3^2 = 7^2 - 4P. 9 = 49 - 4P. 4P = 40, so P = 10.

AI explanation

Using the relationship between roots and coefficients for x2 - 7x + P = 0, the sum of the roots a + b = 7 and the product of the roots ab = P. Since the difference a - b = 3, we square both the sum and the difference to use the algebraic identity (a + b)2 - (a - b)2 = 4ab. Substituting the known values gives 72 - 32 = 4P, which simplifies to 49 - 9 = 4P. Solving for P gives 40 = 4P, resulting in P = 10.

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