Question

Use the Zero Product Property in following exercises, solve.

$3m(2m-5)(m+6)=0$

Short answer

The solutions for the given expression are$m=0,\frac{5}{2}and-6$.

Step-by-step solution

Step 1. Given and explanation.

We have $3m(2m-5)(m+6)=0$.

We know that the zero product property states that if$a\times b=0$, then either of the factors, both are equal to zero.

So putting them equal to zero will give us the values.

Step 2. Evaluating the values. 

We have $3m(2m-5)(m+6)=0$.

Putting them equal to zero gives us,

$$\begin{gathered} 3m=0 \\ m=0 \end{gathered}$$

$$\begin{gathered} 2m-5=0 \\ 2m=5 \\ m=\frac{5}{2} \end{gathered}$$

$$\begin{gathered} m+6=0 \\ m=-6 \end{gathered}$$

This shows that values are $0,\frac{5}{2}and-6$.

Step 3. Check the solution.

We will check the solution by putting the value of the solution in the equation separately. If the equation satisfies, our answer is right.

1. When $m=0$
   $$\begin{gathered} \left[3\left(0\right)\right]\left[2\left(0\right)-5\right]\left[\left(0\right)+6\right]=0 \\ \left(0\right)(0-5)\left(6\right)=0 \\ 0=0 \end{gathered}$$
   Hence satisfied.
2. When $m=\frac{5}{2}$
   $$\begin{gathered} \left[3\left(\frac{5}{2}\right)\right]\left[2\left(\frac{5}{2}\right)-5\right]\left[\left(\frac{5}{2}\right)+6\right]=0 \\ \left(\frac{15}{2}\right)(5-5)\left(\frac{17}{6}\right)=0 \\ \left(\frac{15}{2}\right)\left(0\right)\left(\frac{17}{6}\right)=0 \\ 0=0 \end{gathered}$$
   Hence satisfied.
3. When $m=-6$
   $$\begin{gathered} \left[3(-6)\right]\left[2(-6)-5\right]\left[(-6)+6\right]=0 \\ (-18)(-12-5)\left(0\right)=0 \\ (-18)(-17)\left(0\right)=0 \\ 0=0 \end{gathered}$$
   Hence satisfied.

Thus our calculations are right.