The roots α \alpha α , β \beta β of the equation 3 x 2 + λ x − 1 = 0 3x^2 + \lambda x - 1 = 0 3 x 2 + λ x − 1 = 0 , satisfy 1 α 2 + 1 β 2 = 15. \frac{...

If the equations $$x^2 + mx + 9 = 0$$, $$x^2 + nx + 17 = 0$$, and $$x^2 + (m+n)x + 35 = 0$$ have a c...

The expression $$ a^2 + ab + b^2 $$ is ________ for $$ a < 0 $$ , $$ b < 0 $$. ...

A quadratic equation $$x^2 + bx + 12 = 0$$ has 2 as one of its roots, and another quadratic equatio...

Two numbers, $$a$$ and $$b$$, are such that the quadratic equation $$a x^2 + 3x + 2b = 0$$ has root...

If $$x$$ is a real number, find the smallest value of the expression $$3x^2 - 4x + 7$$. ...

The quadratic function $$ f(x) = ax^2 + bx + c $$, where $$ a < 0 $$, has two distinct roots select...