Multiple choice

The equation ${ ax }^{ 2 }+bx+c=0,{ bx }^{ 2 }+cx+a=0$ have a common root then $\dfrac { { a }^{ 3 }+{ b }^{ 3 }+{ c }^{ 3 } }{ abc } $

  1. 1

  2. 3

  3. 2

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

If the quadratic equations ax^2 + bx + c = 0 and bx^2 + cx + a = 0 share a common root, then either a + b + c = 0 or a = b = c. In both mathematical scenarios, the algebraic identity a^3 + b^3 + c^3 is equal to 3abc. Dividing both sides by abc yields a value of 3.

AI explanation

Let alpha be the common root for the equations ax^2 + bx + c = 0 and bx^2 + cx + a = 0. Substituting alpha into both equations, multiplying the first by b and the second by a, and subtracting them yields alpha(b^2 - ac) = 0. Since a and b are not identical across all coefficients, alpha cannot be zero, so we must have a^3 + b^3 + c^3 = 3abc using the identity for when b^2 = ac. Dividing both sides by abc gives the value of (a^3 + b^3 + c^3) / abc as 3.

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