Question

Find and correct errors of the following mathematical expression:
${\left( {3x + 2} \right)^2} = 3{x^2} + 6x + 4$.

Answer 1

Hint: In the given problem, to find errors first we will simplify the LHS term and compare it with the RHS term. For this, we will use the expansion of ${\left( {a + b} \right)^2}$ which is given by ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.

Complete step by step answer:
In this problem, the mathematical expression ${\left( {3x + 2} \right)^2} = 3{x^2} + 6x + 4 \cdots \cdots \left( 1 \right)$ is given.
We have to find and correct the errors in this expression.
For this, first we will simplify the LHS term. The LHS term of equation $\left( 1 \right)$ is ${\left( {3x + 2} \right)^2}$.
Now we will use the expansion of ${\left( {a + b} \right)^2}$ which is given by ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
So, by using this formula we can write ${\left( {3x + 2} \right)^2} = {\left( {3x} \right)^2} + 2\left( {3x} \right)\left( 2 \right) + {\left( 2 \right)^2}$.
Now we are going to use the law of exponent in the first term. The law of exponent is ${\left( {ab} \right)^m} = {a^m}{b^m}$. Therefore, we can write
${\left( {3x + 2} \right)^2} = {\left( 3 \right)^2}{\left( x \right)^2} + 2\left( {3x} \right)\left( 2 \right) + {\left( 2 \right)^2}$
$\Rightarrow {\left( {3x + 2} \right)^2} = 9{x^2} + 12x + 4$
From the equation $\left( 1 \right)$, we can say that the RHS term is $3{x^2} + 6x + 4$. Now we have the LHS term is $9{x^2} + 12x + 4$ and the RHS term is $3{x^2} + 6x + 4$. So, we can say that LHS is not equal to RHS. If the RHS term of a given expression is $9{x^2} + 12x + 4$ then we can say that given expression is true.

Therefore, the correct mathematical expression is ${\left( {3x + 2} \right)^2} = 9{x^2} + 12x + 4$.

Note: Expansion of ${\left( {a - b} \right)^2}$ is given by ${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$. A mathematical expression must be well-defined. It is not necessary that a mathematical expression must include variables. For example, $5 - 4 = 1$ is a mathematical expression and we can see that there is no variable in this expression. Also $5 - 4 = 2$ is mathematical expression but we can say that it is not a valid (correct) mathematical expression as LHS is not equal to RHS. If there are a minimum two terms and one mathematical operation in the expression then we can say that it is a mathematical expression. In the given problem, $x$ is the variable and variable usually denoted by letter.