2.2.15 Discriminant and Roots

A very popular question is determining the value of an undefined coefficient so that the roots are distinct, equal, or complex.

The roots of a general polynomial $ax^2 + bx + c = 0$ can be determined from the quadratic formula: $r\_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The discriminant is $D = b^2 - 4ac$ .

Example: Find $k$ such that $3x^2 - x - 2k = 0$ has equal roots. $D = (-1)^2 - 4(3)(-2k) = 0 \Rightarrow 1 + 24k = 0 \Rightarrow k = -1/24$

Problem Set 2.2.15

For 2x² - 4x - k = 0 to have 2 equal roots, the smallest value of k is

For 3x² - x - 2k = 0 to have equal roots k has to be

For 3x² - 2x + 1 - k = 0 to have equal roots, k has to be

The discriminant of 2x² - 3x = 1 is

For what value of k does 3x² + 4x + k = 0 have equal roots

For x² - 2x - 3k = 0 to have one real solution k has to be