Question Type 4: For a quadratic dependent on a parameter, finding the value of the parameter that results in either situation of roots Exercises

Find the range of $k$ so that both roots of $x^2 - 2(k+1)x + k = 0$ are less than $1$.

Find the value of $k$ for which the quadratic $x^2 - (k+1)x + k = 0$ has equal roots.

Determine all real $k$ such that the quadratic $kx^2 + 4x + 1 = 0$ has no real roots.

Determine all real values of $k$ for which the quadratic $3kx^2 + 2x + k = 0$ has no real roots.

Find all real $k$ such that the quadratic $(k-1)x^2 + 2kx + 1 = 0$ has one positive root and one negative root.

Find all $k$ for which both roots of $x^2 + (k-3)x + k = 0$ are positive.

Find the value(s) of $k$ such that the quadratic $3kx^2 + 2x + k = 0$ has equal roots.

Find the range of $k$ so that the quadratic $3kx^2 + 2x + k = 0$ has two distinct real roots.

Find $k$ so that the roots of $x^2 + kx + k = 0$ sum to $5$.

Determine $k$ for which the quadratic $2x^2 - kx + 8 = 0$ has equal roots.

Find the non-zero value of $k$ such that one root of $x^2 - kx + k = 0$ is twice the other.

Find all $k$ such that both roots of the quadratic $3kx^2 + 2x + k = 0$ are positive.