Question

How do you solve \[3{{k}^{2}}+72k=33k\] by factoring?

Answer 1

Hint: In this problem, we have to find the value of k by solving the given equation by factoring. We can first subtract the number 33k on both sides, so that we can have the terms on the left-hand side to get simplified. We can then take common terms outside the equation so that we can get the factors from which we can find the value of k.

Complete step by step answer:
We know that the given equation to be solved is,
 \[3{{k}^{2}}+72k=33k\]
We can now subtract the number 33k on both sides, we get
\[\Rightarrow 3{{k}^{2}}+72k-33k=33k-33k\]
Now we can simplify the above step by cancelling the similar terms with opposite sign in the right-hand side, we get
\[\Rightarrow 3{{k}^{2}}+39k=0\]
We can see that we have 3k as a common factor in both the terms, so that we can take outside the equation, we get
\[\Rightarrow 3k\left( k+13 \right)=0\]
We can see that the above terms are the factors of the given equation.
We can now separate the above two terms and we can individually equate the terms to 0, we get
\[\begin{align}
  & \Rightarrow 3k=0,k+13=0 \\
 & \Rightarrow k=0,-13 \\
\end{align}\]
Therefore, the value of \[k=0,-13\].

Note:
Students make mistakes while finding the factors from the given equation by taking the common terms outside the equation. We are given that we have to find the value of k by factoring, so we have to find the factors from which we can solve and find the value of k. We can now substitute the resulting value in any of the equations.
We can substitute the values \[k=0,-13\] in the equation \[3{{k}^{2}}+72k=33k\],
When k = -13,
\[\begin{align}
  & \Rightarrow 3{{\left( -13 \right)}^{2}}+72\left( -13 \right)=33\left( -13 \right) \\
 & \Rightarrow 507-936=-429 \\
\end{align}\]
When k = 0,
\[\begin{align}
  & \Rightarrow 3\left( 0 \right)+72\left( 0 \right)=33\left( 0 \right) \\
 & \Rightarrow 0=0 \\
\end{align}\]
Therefore, the values are correct.