Question

If $ a, b $ are roots of the equation $ x^2 - 5x + 6 = 0 $, find the value of $ a^3 + b^3 $.

• A. 125
• B. 215
• C. 98
• D. 35

Hint:

Use the identity \( a^3 + b^3 = (a + b)^3 - 3ab(a + b) \) or directly compute if roots are simple.

Answer 1

The Correct Option is D

Step 1: Find the roots of the quadratic equation. The equation is: \[ x^2 - 5x + 6 = 0 \] Factoring the equation yields: \[ x^2 - 5x + 6 = (x - 2)(x - 3) \Rightarrow \text{roots are } a = 2, \quad b = 3 \] Step 2: Calculate the cube of each root. \[ a^3 = 2^3 = 8, \quad b^3 = 3^3 = 27 \] Step 3: Sum the calculated cubes. \[ a^3 + b^3 = 8 + 27 = \boxed{35} \]