Question

Simplify the expression \[{(2x + 5)^2} - {(2x - 5)^2}\]

Answer 1

Hint: There are algebraic identities in this question, and we will apply the formulas for algebraic identities to derive the appropriate solution for the given question by simplifying the terms. The two algebraic identities that we apply here are \[{(a + b)^2} = {a^2} + {b^2} + 2ab\] and \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]

Complete step-by-step solution:
An algebraic expression is made up of constants, variables, and arithmetic operation symbols. There are three types of algebraic expressions: monomial, binomial, and polynomial. We have certain algebraic identities for various algebraic expressions.
So, we have been given the equation, that is
\[{(2x + 5)^2} - {(2x - 5)^2}\]
This is a question in which the subtraction of the two algebraic identities is required. Here are the algebraic identities we'll use:
\[{(a + b)^2} = {a^2} + {b^2} + 2ab\] and
\[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
These Identities were provided to open the bracket that is squared or raised to a power of two.
Here, we will take,
\[a = 2x\] and \[b = 5\]
Therefore, by putting the formula, the equation will become
\[{(2x + 5)^2} = {(2x)^2} + {(5)^2} + (2 \times 2x \times 5)\] and
\[{(2x - 5)^2} = {(2x)^2} + {(5)^2} - (2 \times 2x \times 5)\]
By solving them separately, we will get
\[ \Rightarrow {(2x + 5)^2} = {2^2}{x^2} + {5^2} + (2 \times 10x)\] and \[{(2x - 5)^2} = {2^2}{x^2} + {5^2} - (2 \times 10x)\]
\[ \Rightarrow {(2x + 5)^2} = 4{x^2} + 25 + 20x\] and \[{(2x - 5)^2} = 4{x^2} + 25 - 20x\]
These expressions cannot be further simplified because each term is different and hence, we will now apply these expressions in the statement given above.
Now, by subtracting the second term from the first term to derive the equation
\[ \Rightarrow {(2x + 5)^2} - {(2x - 5)^2} = (4{x^2} + 25 + 20x) - (4{x^2} + 25 - 20x)\]
Simplifying the equation by opening the brackets, we get
\[ \Rightarrow {(2x + 5)^2} - {(2x - 5)^2} = 4{x^2} + 25 + 20x - 4{x^2} - 25 + 20x\]
Putting similar terms together, we get
\[ \Rightarrow {(2x + 5)^2} - {(2x - 5)^2} = 4{x^2} - 4{x^2} + 20x + 20x + 25 - 25\]
Now, cancelling the terms,
\[ \Rightarrow {(2x + 5)^2} - {(2x - 5)^2} = 40x\]
Hence, the simplified form of the given equation is \[40x\].

Note: Remember to apply the square in both the variable and the constant while opening the brackets. Also, don’t forget to consider - sign in all the terms which are under the bracket. The difference between the two expressions is + and – which will further differentiate the formula by putting \[ + 2ab\] and \[ - 2ab\] respectively. We should also know the basic algebraic identities to get the correct solution.