If p and q are the roots of the equation 2x2 − x − 4 = 0. Find p3 + q3

Sure, let's solve this step by step.

Step 1: Identify the coefficients of the quadratic equation. The quadratic equation is 2x^2 - x - 4 = 0. Here, a = 2, b = -1, and c = -4.

Step 2: Use the formula for roots of a quadratic equation. The roots of a quadratic equation ax^2 + bx + c = 0 are given by the formula p, q = [-b ± sqrt(b^2 - 4ac)] / 2a.

Step 3: Identify the roots as p and q. We don't need to calculate the actual roots for this problem. We just need to know that p and q are the roots of the equation.

Step 4: Use the identity for the sum of cubes. The identity for the sum of cubes is p^3 + q^3 = (p + q)(p^2 - pq + q^2).

Step 5: Use the sum and product of roots. From the quadratic equation, we know that p + q = -b/a = 1/2 and pq = c/a = -4/2 = -2.

Step 6: Substitute these values into the identity. p^3 + q^3 = (1/2)( (1/2)^2 - (1/2)(-2) + (-2)^2 ) = (1/2)(1/4 + 1 + 4) = (1/2)(5.25) = 2.625.

So, p^3 + q^3 = 2.625.