Quadratic Equations and Inequalities: Practice and Methods

Factor the left-hand side.

Hence

With discriminant $D$,

$D = 0$ gives $m = -2,\ 5$. Since $m > 0$,

The double root is then

Factoring the left side,

If $a > 0$, the solution is $0 < x < a$. The integer solutions are exactly $1,\ 2$ when

If $a < 0$, the solution is $a < x < 0$. The integer solutions are exactly $-2,\ -1$ when

If $a = 0$ there is no solution. Therefore

Factor the left-hand side.

The roots of $(x - 5)(x + 2) = 0$ are $x = -2,\ 5$, so

With discriminant $D$,

The condition is $D > 0$, so

Let $f(x) = x^2 - 2mx - 2m$; its graph opens upward with axis $x = m$. The conditions are

The discriminant gives

The value at the point is

Combining the conditions,