Multiple choice

Directions: Answer the question independently. The roots of the equations x2 + bx + c = 0 are equal to that of equation bx2 + cx + 8 = 0. The roots of the equation x3 - 9x2 + 15x - 7 = 0 are 'a' and 'd' with a > d. What is the value of a - b + c + d? (Given b, c > 0)

  1. -1

  2. 14

  3. 4

  4. 10
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

For the equations x^2 + bx + c = 0 and bx^2 + cx + 8 = 0 to have equal roots, the ratios of coefficients must be equal: 1/b = b/c = c/8. This implies b^2 = c and c^2 = 8b. Solving these yields b = 2 and c = 4. The roots of x^3 - 9x^2 + 15x - 7 = 0 are found by testing small integers; x=1 is a root, leading to (x-1)(x^2 - 8x + 7) = 0, which factors to (x-1)(x-1)(x-7) = 0. Thus, a=7 and d=1. The value of a - b + c + d is 7 - 2 + 4 + 1 = 10.

AI explanation

Let r1 and r2 be the common roots of x2 + bx + c = 0 and bx2 + cx + 8 = 0. Equating the ratios of corresponding coefficients gives c/b = 8/c and 1/b = c/8. Solving c2 = 8b and b c = 8 yields b = 2 and c = 4. The equation x3 - 9x2 + 15x - 7 = 0 has roots a, d, and 2. Substituting x = 2 gives 8 - 36 + 30 - 7 = -5, which is not a root. Let a and d be the roots of the quadratic factor (x3 - 9x2 + 15x - 7)/(x - 1), so (x - 1)(x2 - 8x + 7) = 0. Since a > d, the roots of x2 - 8x + 7 = 0 give a = 7 and d = 1. The value of a - b + c + d is 7 - 2 + 4 + 1 = 10.

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