1. **State the problem:** Simplify and understand the expression for $J$ given by $$J=\frac{-(d - m_2) \pm \sqrt{(d - m_2)^2 - 4 \left(\frac{1}{k}\right) \left(b \times \frac{y m_3}{\alpha (c - m_3 h)} \div (m_1 + e) \right)}}{2 \left(\frac{1}{k}\right)}$$ 2. **Simplify the denominator:** The denominator is $2 \times \frac{1}{k}$, which equals $\frac{2}{k}$. 3. **Rewrite the equation:** $$J = \frac{-(d - m_2) \pm \sqrt{(d - m_2)^2 - 4 \left(\frac{1}{k}\right) \left(b \times \frac{y m_3}{\alpha (c - m_3 h)} \div (m_1 + e) \right)}}{\frac{2}{k}}$$ 4. **Simplify division by fraction:** Dividing by $\frac{2}{k}$ is equivalent to multiplying by $\frac{k}{2}$, so $$J = \left(-(d - m_2) \pm \sqrt{(d - m_2)^2 - 4 \times \frac{1}{k} \times \frac{b y m_3}{\alpha (c - m_3 h)(m_1 + e)}} \right) \times \frac{k}{2}$$ 5. **Simplify inside the square root:** Calculate $$4 \times \frac{1}{k} \times \frac{b y m_3}{\alpha (c - m_3 h)(m_1 + e)} = \frac{4 b y m_3}{k \alpha (c - m_3 h)(m_1 + e)}$$ 6. **Final simplified expression:** $$J = \frac{k}{2} \left( - (d - m_2) \pm \sqrt{(d - m_2)^2 - \frac{4 b y m_3}{k \alpha (c - m_3 h)(m_1 + e)}} \right)$$ This form expresses $J$ clearly in terms of the variables and constants. This completes the simplification and explanation.