Question

Find the value of \[94 \times 106\].

Answer 1

Hint: Here, we need to find the value of \[94 \times 106\] using a suitable standard identity. We will rewrite the two numbers as a sum or difference of two numbers in such a way that the product can be found using one of the standard algebraic identities. Then, we will simplify the expression and find the required value.

Formula Used:
The product of the sum of two number and the difference of two numbers can be calculated using the algebraic identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\].

Complete step-by-step answer:
We can evaluate the given product using any of the two identities \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\] or \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\].
We will use the second identity to solve this problem.
First, we will rewrite the given numbers as the sum or difference of two numbers such that the identity is applicable.
We know that 94 is the difference of 100 and 6, and 106 is the sum of 100 and 6.
Therefore, substituting \[94 = 100 - 6\] and \[106 = 100 + 6\], we can rewrite the product as
\[ \Rightarrow 94 \times 106 = \left( {100 - 6} \right)\left( {100 + 6} \right)\]
Substituting \[a = 100\] and \[b = 6\] in the algebraic identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\], we get
\[ \Rightarrow \left( {100 - 6} \right)\left( {100 + 6} \right) = {100^2} - {6^2}\]
Simplifying the expression, we get
\[ \Rightarrow 94 \times 106 = 10000 - 36\]
Subtracting 36 from 10000, we get
\[ \Rightarrow 94 \times 106 = 9964\]
\[\therefore \] The value of the product \[94 \times 106\] is 9964.

Note: We can also solve the problem using the identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\].
Substituting \[94 = 90 + 4\] and \[106 = 90 + 16\], we can rewrite the product as
\[ \Rightarrow 94 \times 106 = \left( {90 + 4} \right)\left( {90 + 16} \right)\]
Now, we will use the algebraic identity \[\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab\].
Substituting \[x = 90\],\[a = 4\] and \[b = 16\] in the algebraic identity, we get
\[ \Rightarrow \left( {90 + 4} \right)\left( {90 + 16} \right) = {90^2} + \left( {4 + 16} \right)90 + \left( 4 \right)\left( {16} \right)\]
Simplifying the expression, we get
\[\begin{array}{l} \Rightarrow 94 \times 106 = 8100 + 20 \times 90 + 64\\ \Rightarrow 94 \times 106 = 8100 + 1800 + 64\end{array}\]
Adding the terms of the expression, we get
\[ \Rightarrow 94 \times 106 = 9964\]
\[\therefore \] The value of the product \[94 \times 106\] is 9964.