Question

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

Answer 1

Hint: In the given problem, to find errors first we will simplify the LHS term and compare it with the RHS term. To simplify $\left( {2a + 3b} \right)\left( {a - b} \right)$, we will multiply each term of the first bracket with each term of the second bracket.

Complete step by step answer:
In this problem, the mathematical expression $\left( {2a + 3b} \right)\left( {a - b} \right) = 2{a^2} - 3{b^2} \cdots \cdots \left( 1 \right)$ 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 the equation $\left( 1 \right)$ is $\left( {2a + 3b} \right)\left( {a - b} \right)$.
Now we will simplify this by multiplying each term of the first bracket with each term of the second bracket. So, we can write $\left( {2a + 3b} \right)\left( {a - b} \right) = \left( {2a} \right)\left( a \right) - \left( {2a} \right)\left( b \right) + \left( {3b} \right)\left( a \right) - \left( {3b} \right)\left( b \right)$. Let us simplify this equation. Therefore, we can write
$\left( {2a + 3b} \right)\left( {a - b} \right) = 2{a^2} - 2ab + 3ba - 3{b^2} \\
\Rightarrow \left( {2a + 3b} \right)\left( {a - b} \right) = 2{a^2} - 2ab + 3ab - 3{b^2} \\
\Rightarrow \left( {2a + 3b} \right)\left( {a - b} \right) = 2{a^2} + ab - 3{b^2} \\$
From the equation $\left( 1 \right)$, we can say that the RHS term is $2{a^2} - 3{b^2}$. Now we have the LHS term is $2{a^2} + ab - 3{b^2}$ and the RHS term is $2{a^2} - 3{b^2}$. So, we can say that LHS is not equal to RHS. If the RHS term of a given expression is $2{a^2} + ab - 3{b^2}$ then we can say that given expression is true.

Therefore, the correct mathematical expression is $\left( {2a + 3b} \right)\left( {a - b} \right) = 2{a^2} + ab - 3{b^2}$.

Note: In the given problem, we can use the identity $\left( {x + p} \right)\left( {x - q} \right) = {x^2} + \left( {p - q} \right)x - pq$. A mathematical expression must be well-defined. It is not necessary that a mathematical expression must include variables. For example, $16 \div 4 = 4$ is a mathematical expression and we can see that there is no variable in this expression. Also $16 \div 4 = - 4$ 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 of two terms and one mathematical operation in the expression then we can say that it is a mathematical expression.