Question

What is the product of \[{\left( {2x - 5} \right)^2}\]?

Answer 1

Hint: For solving this equation first of all we should be aware of the term product. The term product is generally referred to as the result of one or more multiplications. We can also say that, when we multiply two terms together then after multiplication we obtain a quantity or number that is generally referred to as the product of two terms.

Complete step by step answer:
About multiplication:-
The repeated addition of one number to the number of times equal to the other number is known as the multiplication of two numbers. It is also known as the mathematical operation which is performed on a pair of numbers in order to get a third number known as a product.
The properties of multiplication are – distributive, associative and commutative.

Distributive property of Multiplication –The sum of the multiplication of the number by each of the amount is just equal to the multiplication of a number by its sum.
For example - $2\,\, \times \,\,\left( {3 + 5} \right)$ is $2\,\, \times \,\,\left( {3 + 5} \right)\,\, = \,\,\,2 \times 3\,\, + \,\,2 \times 5$

Commutative property: The product does not change by the order of the number.
For example - $10\,\, \times \,\,3\,\, = \,\,3\,\, \times \,\,10$

Associative property: The result of the multiplication does not change the mode of grouping.
For example - $\left( {3\,\, \times \,\,2} \right)\,\, \times \,\,5 = \,3\,\, \times \,\,\left( {2\,\, \times \,\,5} \right)$

Now, according to Question:-
From the given question we have to obtain the product of \[{\left( {2x - 5} \right)^2}\]
Thus, by using the identity ${\left( {a - b} \right)^2} = \,\,{a^2} + \,\,{b^2} - \,\,2ab$
Hence, from the question we can write
 \[{\left( {2x - 5} \right)^2}\, = \,\,\,4{x^2} + \,25 - 2\left( {2x\,\, \times \,\,5} \right)\]
 Or $4{x^2} + \,\,25\,\, - \,\,20x$

Thus, the product of the given equation \[{\left( {2x - 5} \right)^2}\] is $4{x^2} + \,\,25\,\, - \,\,20x$.

Note: The term factor and product both are different. The product is just the result of the multiplication of numbers while factors are the simple parts of a number when multiplied together we obtain the original number.