Question

How do you solve the \[3b\left( 9b-27 \right)=0\]?

Answer 1

Hint: This type of problem is based on the concept of solving quadratic equations with variable b. First, we have to consider the whole function and then take 3 common forms (9b-27). Then, we need to divide the whole quadratic equation by 27 and simplify the given equation. Thus, we find the factors of the given equation by making necessary calculations. And then solve b by making some adjustments to the obtained factors of the equation and equating then to zero.

Complete step-by-step answer:
According to the question, we are asked to solve the given quadratic equation\[3b\left( 9b-27 \right)=0\].
We have been given the equation is \[3b\left( 9b-27 \right)=0\]. -----(1)
Let us first consider equation (1).
We first have the common terms from the left-hand side of the quadratic equation (1).
\[\Rightarrow b=0\left( b-3 \right)=0\]
Here, 9 is the common term.
Therefore, we get,
\[9\left( 3b \right)\left( b-3 \right)=0\]
On further simplifications, we get,
\[\Rightarrow \left( 9\times 3b \right)\left( b-3 \right)=0\]
\[\Rightarrow \left( 27b \right)\left( b-3 \right)=0\] -----(2)
Now, let us divide the whole equation (2) by 27.
We get,
\[\dfrac{27b\left( b-3 \right)}{27}=\dfrac{0}{27}\]
We know that 0 divided by any term is zero.
Therefore,
\[\dfrac{27b\left( b-3 \right)}{27}=0\]
On further simplifications, we get,
\[\Rightarrow b\left( b-3 \right)=0\]
Thus, the factors of the quadratic equation are b and (b-3).
We know that the factors are always equal to zero.
\[\Rightarrow b=0\] and \[\left( b-3 \right)=0\]
Let us now add 3 on both the sides of \[\left( b-3 \right)=0\].
We get,
b-3+3=3
Since same term with opposite sign cancels out, we get,
b=3
Therefore, the values of b are 0 and 3.

Hence, the values of b in the equation \[3b\left( 9b-27 \right)=0\] are b=0 and b=3.

Note: Whenever you get this type of problem, we should always try to make the necessary changes in the given inequality to get the final solution of the inequality which will be the required answer. We should avoid calculation mistakes based on sign conventions. We can also solve this problem by directly dividing the whole quadratic equation by 3 in the first step and then follow the same steps mentioned above.