The equation $2x^2 - 3x - (k + 1) = 0$, where $k$ is a constant, has no real roots - Edexcel - A-Level Maths Pure - Question 7 - 2007 - Paper 2

For the quadratic equation in the form $ax^2 + bx + c = 0$, the discriminant $ext{D}$ is given by $D = b^2 - 4ac$. In this case, we have:

• $a = 2$,
• $b = -3$,
• $c = -(k + 1)$.
  We calculate the discriminant:

$D = (-3)^2 - 4(2)(-(k + 1)) = 9 + 8(k + 1) = 9 + 8k + 8 = 8k + 17$

For the equation to have no real roots, the discriminant must be less than zero:

$8k + 17 < 0$