If $x=3$ and $x=2$ are the roots of the equation $\dfrac { 1 } { 2 } \left( 2 x ^ { 2 } - 10 x + k \right) = 4 ,$ then find the value of 0.5 k ?

Using the sum of roots formula for the quadratic equation 1/2(2x^2 - 10x + k) = 4, which simplifies to x^2 - 5x + k/2 - 4 = 0, we know the sum of the roots (3 + 2) equals 5. This matches the coefficient of x, confirming the structure, so we use the product of the roots. The product of the roots (3 multiplied by 2) equals k/2 - 4, giving 6 = k/2 - 4. Solving this gives k/2 = 10, which means 0.5 k is 10.

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