Multiple choice

If α,β,γ are the roots of the equation 2x3 -3x2 +6x+1 = 0, then α2 + β2 + γ2 is equal to

  1. 21/2

  2. -33/4

  3. -15/4

  4. 5
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

For 2x^3 - 3x^2 + 6x + 1 = 0, sum of roots (a+b+c) = 3/2, sum of roots taken two at a time (ab+bc+ca) = 6/2 = 3. a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) = (1.5)^2 - 2(3) = 2.25 - 6 = -3.75 = -15/4.

AI explanation

Let the roots of the cubic equation ax3 + bx2 + cx + d = 0 be α, β, and γ; the sum of the roots is -b/a and the sum of the product of the roots taken two at a time is c/a. For 2x3 - 3x2 + 6x + 1 = 0, the sum α + β + γ equals -(-3)/2 = 3/2, and the sum αβ + βγ + γα equals 6/2 = 3. The identity for the sum of the squares of the roots is α2 + β2 + γ2 = (α + β + γ)2 - 2(αβ + βγ + γα). Substituting the values gives (3/2)2 - 2(3), which is 9/4 - 6, resulting in -15/4.

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