Question

Find the value of given expression ${\left( {b - 7} \right)^2}$

Answer 1

Hint: The given expression ${\left( {b - 7} \right)^2}$ consists of the variable ‘b’, which can be treated as any number and can be operated like the same. Use the algebraic identity of the square of the difference, i.e. ${\left( {x - y} \right)^2} = {x^2} - 2xy + {y^2}$ . For getting the required expansion put $x = b$ and $y = 7$ , then solve it further.

Complete step-by-step answer:
Here in this problem, we are given with an expression ${\left( {b - 7} \right)^2}$ , which is in the form of a square of a linear expression. And using the algebraic properties we need to expand this expression.
In the expression ${\left( {b - 7} \right)^2}$ , ‘b’ here is a variable that does not have a constant value and $7$ which is a real number with a constant value. In mathematics, a variable is a symbol which functions as a placeholder for varying expressions or quantities and is often used to represent an arbitrary element of a set.
The expansion of the given expression can be done using an algebraic identity. The algebraic equations which are valid for all values of variables in them are called algebraic identities. They are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and solving different polynomials.
We know the algebraic identity ${\left( {x - y} \right)^2} = {x^2} - 2xy + {y^2}$ , if we put $x = b$ and $y = 7$ then we will get:
$ \Rightarrow {\left( {b - 7} \right)^2} = {b^2} - 2 \times b \times 7 + {7^2}$
Now we can solve these terms by using the square of $7$ as ${7^2} = 49$
$ \Rightarrow {\left( {b - 7} \right)^2} = {b^2} - 2 \times b \times 7 + {7^2} = {b^2} - 14b + 49$
Therefore, the expansion of the expression ${\left( {b - 7} \right)^2}$ is given by ${\left( {b - 7} \right)^2} = {b^2} - 14b + 49$
Additional Information: To understand the concept of expansion better, we can use another example where we can expanding the expression ${\left( {m + 3} \right)^2}$
For this we can use the algebraic identity ${\left( {x + y} \right)^2} = {x^2} + 2xy + {y^2}$ , where we can put $x = m$ and $y = 3$ , this will give us:
$ \Rightarrow {\left( {m + 3} \right)^2} = {m^2} + 2 \times m \times 3 + {3^2}$
This can be further simplified by using ${3^2} = 9$
$ \Rightarrow {\left( {m + 3} \right)^2} = {m^2} + 2 \times m \times 3 + {3^2} = {m^2} + 6m + 9$
Therefore, for the expression, ${\left( {m + 3} \right)^2}$ we got the expansion as ${\left( {m + 3} \right)^2} = {m^2} + 6m + 9$.

Note: In questions like this the use of algebraic identities plays a crucial role in the solution. An alternative approach to the same problem can be taken by multiplying the expression to itself to find the square, i.e. ${\left( {b - 7} \right)^2} = \left( {b - 7} \right) \times \left( {b - 7} \right) = b\left( {b - 7} \right) - 7\left( {b - 7} \right) = {b^2} - 7b - 7b + 7 \times 7$ . This will also give us the same expanded expression as ${\left( {b - 7} \right)^2} = {b^2} - 14b + 49$ . This method does not include any use of identity but uses the distributive property.