Number of real values of $\lambda$ so the equation ${x}^{2}-3x+2\lambda=0$ and ${x}^{2}-4\lambda x+3=0$ has exactly one root common

Let alpha be the single common root of the equations x squared minus 3 x plus 2 lambda equals 0 and x squared minus 4 lambda x plus 3 equals 0. Subtracting the second equation from the first gives 4 lambda x minus 3 x plus 2 lambda minus 3 equals 0, which factors to x plus 1 times 4 lambda minus 3 equals 0. Since they have exactly one common root and lambda cannot be three fourths, the common root alpha must be negative 1. Substituting x equals negative 1 into the first equation gives 1 plus 3 plus 2 lambda equals 0, so lambda equals negative 2. Checking lambda equals negative 2 in the second equation gives x squared plus 8 x plus 3 equals 0, which yields roots of negative 1 and negative 7, confirming exactly one common root. Thus, there is exactly 1 real value of lambda.

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