Question

The equation kx^2 + 3x + 1=0 has exactly one solution. Find the value of k.

Answer 1

we have to require that the discriminat of such trinomial, is zero, namely:
\[\huge \Delta = 9 - 4k = 0\]

Answer 2

so you need to know when in what case of discriminant one quadratic has one solution ?

Answer 3

standard form of a quadratic and its solution are
\[ax^2 + bx + c = 0\]\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]to have only one solution, the term under the radical (aka the discriminant) must be \(0\)

Viewed a different way, the vertex of the parabola will be on the x-axis, which is the only way a quadratic can have a single solution.

with our equation, we have \(a=k\) and \(b=3\), so the value of \(x\) that solves the equation will be \[x = -\frac{3}{2k}\]

Note that this is NOT the answer to your question! However, it can be used to check your answer for correctness.