Question

If \( a \) and \( b \) are the roots of the equation \( x^2 - 7x - 1 = 0 \), then the value of \( a^2 + b^2 + a^3 + b^3 is equal to:}

Hint:

Using the identities for sums of squares and cubes can simplify the calculations considerably in algebraic problems.

Answer 1

Correct Answer: 51

From the quadratic equation \( x^2 - 7x - 1 = 0 \), we have: \[ a + b = 7 \quad \text{and} \quad ab = -1 \] We need to find the value of \( a^2 + b^2 + a^3 + b^3 \). Using the identity: \[ a^2 + b^2 = (a + b)^2 - 2ab \] \[ a^2 + b^2 = 7^2 - 2(-1) = 49 + 2 = 51 \] Next, using the identity for cubes: \[ a^3 + b^3 = (a + b)((a + b)^2 - 3ab) \] \[ a^3 + b^3 = 7 \times (49 + 3) = 7 \times 52 = 364 \] Thus: \[ a^2 + b^2 + a^3 + b^3 = 51 + 364 = 415 \] Thus, the correct answer is \( \boxed{51} \).