+95
Let a and b and 2x2 - 8x + 7 = 0 be the roots of the quadratic. Compute a3 + b3
The roots of the quadratic 2x2 - 8x + 7 = 0 are given by the quadratic formula:
x = (8 ± √(8^2 - 4 * 2 * 7)) / (2 * 2) = (8 ± √(64 - 56)) / 4 = (8 ± 2√2) / 4 = 2 ± √2
a3 + b3 = (2 + √2)^3 + (2 - √2)^3 = 8 + 12√2 + 2 + 8 - 12√2 + 2 = 35
Therefore, a^3 + b^3 = 35.
The sum of the roots of the quadratic 2x2 - 8x + 7 = 0 is given by -(-8)/2 = 4. The product of the roots is given by 7/2.
We can compute a3 + b3 as follows:
a3 + b3 = a3 + ab2 + b2a + b3 = (a + b)(a2 - ab + b2) = (4)(a2 - ab + 7/2)
We know that a2 - ab + b2 = (a + b)2 - 2ab = 16 - 8 = 8. Therefore, a3 + b3 = (4)(8) = 32.
Therefore, the answer is 32.
