What are the roots of the equation? \[ \begin{array}{l}\text { Answer } \\ \qquad \begin{array}{l}\frac{3}{2}+10 x=3 x+15 \\ \frac{3}{2} \text { and }-5\end{array} \\ \frac{7 \pm i \sqrt{71}}{4} \\ \text { Suld }-\frac{3}{2} \\ \text { Answer }\end{array} \]

To find the roots of the equation given, we first need to clarify the equation itself. The equation appears to be: \[ \frac{3}{2} + 10x = 3x + 15 \] Let's solve this step by step. 1. **Rearranging the equation**: We want to isolate \(x\). First, we can move all terms involving \(x\) to one side and constant terms to the other side. \[ 10x - 3x = 15 - \frac{3}{2} \] 2. **Simplifying the left side**: \[ 7x = 15 - \frac{3}{2} \] 3. **Calculating the right side**: To perform the subtraction, we need a common denominator. The number 15 can be expressed as \(\frac{30}{2}\): \[ 15 - \frac{3}{2} = \frac{30}{2} - \frac{3}{2} = \frac{27}{2} \] 4. **Substituting back into the equation**: \[ 7x = \frac{27}{2} \] 5. **Dividing both sides by 7**: \[ x = \frac{27}{2 \cdot 7} = \frac{27}{14} \] Now, we have found one root of the equation, which is \(x = \frac{27}{14}\). Next, let's check the other roots mentioned in the problem statement, specifically \(\frac{3}{2}\), \(-5\), and \(\frac{7 \pm i \sqrt{71}}{4}\). Since the equation we solved is linear, it can only have one root. The other values provided do not correspond to roots of the equation we solved. Thus, the only root of the equation \(\frac{3}{2} + 10x = 3x + 15\) is: \[ \boxed{\frac{27}{14}} \] The other values mentioned do not apply to this equation.