Question

${\left( {35} \right)^2} - {\left( {28} \right)^2} = {m^2}$. What is the value of m?
(1) 7
(2)21
(3) 49
(4) 14

Answer 1

The given expression is a quadratic equation. We will use the algebraic identity ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ to simplify the right-hand side of the given expression. To do so, we will compare the given equation with the algebraic identity to find the value of a and b. At last, we will equate both sides to find the value of m.

Complete step by step answer:
The given expression is ${\left( {35} \right)^2} - {\left( {28} \right)^2} = {m^2}$.
We have to find the value of m.
On comparing the right-hand side of the given expression with the right-hand side of algebraic identity${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$, we find:

a = 35\\\ b = 28 \end{array}$$ By using the algebraic identity $${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$$, we can write the given expression as: $$\begin{array}{c} {\left( {35} \right)^2} - {\left( {28} \right)^2} = {m^2}\\\ \left( {35 + 28} \right)\left( {35 - 28} \right) = {m^2} \end{array}$$ On solving the above expression, we can write: $$441 = {m^2}$$ Taking the square root of both the sides, we get: $$\begin{array}{c} \sqrt {441} = \sqrt {{m^2}} \\\ m = 21 \end{array}$$ Therefore, the value of m for the given expression is $$21$$, and option (2) is correct. **Note:** Alternate method: We can solve the given expression without any use of algebraic expression by substituting the values of the square of 35 and 28 in the given expression, but to do so, we have to remember the square values of 35 and 21. It is not convenient to remember the square of these numbers. Also, if we remember them, we can only solve this question, not of its similar type. Therefore, this method will be on the least preference for us to solve similar problems.