Question

Find and correct errors of the following mathematical expression:
${\left( {z + 5} \right)^2} = {z^2} + 25$

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 answer:
In this problem, the mathematical expression ${\left( {z + 5} \right)^2} = {z^2} + 25 \cdots \cdots \left( 1 \right)$ 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 ${\left( {z + 5} \right)^2}$.
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 ${\left( {z + 5} \right)^2} = {\left( z \right)^2} + 2\left( z \right)\left( 5 \right) + {\left( 5 \right)^2}$.
Let us simplify this equation. Therefore, we can write ${\left( {z + 5} \right)^2} = {z^2} + 10z + 25$.
From the equation $\left( 1 \right)$, we can say that the RHS term is ${z^2} + 25$. Now we have the LHS term is ${z^2} + 10z + 25$ and the RHS term is ${z^2} + 25$. So, we can say that LHS is not equal to RHS. If the RHS term of given expression is ${z^2} + 10z + 25$ then we can say that given expression is true.

Therefore, the correct mathematical expression is ${\left( {z + 5} \right)^2} = {z^2} + 10z + 25$.

Note: In the given problem, we can write ${\left( {z + 5} \right)^2} = \left( {z + 5} \right)\left( {z + 5} \right)$. Now we can use the identity $\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$ to find error in the given expression. A mathematical expression must be well-defined. It is not necessary that a mathematical expression must include variables. For example, $5 + 4 = 9$ is a mathematical expression and we can see that there is no variable in this expression. Also $5 + 4 = 1$ 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. In the given problem, $z$ is the variable and variable usually denoted by letter.