Question

If \[a\] and \[b\] are any two real numbers with opposite signs, which of the following is the greatest?
A.\[{\left( {a - b} \right)^2}\]
B.\[{\left( {\left| a \right| - \left| b \right|} \right)^2}\]
C.\[\left| {{a^2} - {b^2}} \right|\]
D.\[{a^2} + {b^2}\]

Answer 1

Hint: Here we will first expand the given algebraic expressions given in the option and we will consider the sign of these numbers to be opposite. Then we will compare the given algebraic expressions to check which one is greater among them.

Complete step-by-step answer:
It is given that \[a\] and \[b\] are any two real numbers with opposite signs, we have to check which of the following algebraic expression formed using these real numbers in the given option is greatest.
We will consider all the algebraic expression one by one.
The first algebraic expression is \[{\left( {a - b} \right)^2}\]
We will first expand this algebraic expression using algebraic identities.
\[ \Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\] ………………. \[\left( 1 \right)\]
As it is given that \[a\] and \[b\] are any two real numbers with opposite signs. We will consider \[a\] to be positive and \[b\] to be negative and now, we will put the signs of the number in the right side of equation \[\left( 1 \right)\].
\[ \Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} - 2a\left( { - b} \right)\]
On simplification, we get
\[ \Rightarrow {\left( {a - b} \right)^2} = {a^2} + {b^2} + 2ab\] …………….. \[\left( 2 \right)\]
The second algebraic expression is \[{\left( {\left| a \right| - \left| b \right|} \right)^2}\].
We will first expand this algebraic expression using algebraic identities.
\[ \Rightarrow {\left( {\left| a \right| - \left| b \right|} \right)^2} = {\left| a \right|^2} + {\left| b \right|^2} - 2 \times \left| a \right| \times \left| b \right|\] ………………. \[\left( 3 \right)\]
As it is given that \[a\] and \[b\] are any two real numbers with opposite signs. We will consider \[a\] to be positive and \[b\] to be negative and now, we will put the signs of the number in the right side of equation \[\left( 3 \right)\].
\[ \Rightarrow {\left( {\left| a \right| - \left| b \right|} \right)^2} = {\left| a \right|^2} + {\left| { - b} \right|^2} - 2 \times \left| a \right| \times \left| { - b} \right|\]
On simplification, we get
\[ \Rightarrow {\left( {\left| a \right| - \left| b \right|} \right)^2} = {a^2} + {b^2} - 2ab\] ………….. \[\left( 4 \right)\]
The third algebraic expression is \[\left| {{a^2} - {b^2}} \right|\]
We can write it as \[{a^2} - {b^2} > 0\] ……………. \[\left( 5 \right)\]
As it is given that \[a\] and \[b\] are any two real numbers with opposite signs. We will consider \[a\] to be positive and \[b\] to be negative and now, we will put the signs of the number in the right side of equation \[\left( 5 \right)\].
\[ \Rightarrow {a^2} - {\left( { - b} \right)^2} > 0\]
On simplification, we get
\[ \Rightarrow {a^2} - {b^2} > 0\] ………….. \[\left( 6 \right)\]
The third algebraic expression is \[{a^2} + {b^2}\]
We can write it as
\[ \Rightarrow {a^2} + {b^2} > 0\]……………… \[\left( 7 \right)\]
As it is given that \[a\] and \[b\] are any two real numbers with opposite signs. We will consider \[a\] to be positive and \[b\] to be negative and now, we will put the signs of the number in the right side of equation \[\left( 7 \right)\].
\[ \Rightarrow {a^2} + {\left( { - b} \right)^2} > 0\]
On further simplification, we get
\[ \Rightarrow {a^2} + {b^2} > 0\] ……………….. \[\left( 8 \right)\]
On comparing equation \[\left( 2 \right)\], equation \[\left( 4 \right)\], equation \[\left( 6 \right)\] and equation \[\left( 8 \right)\], we get
\[{a^2} + {b^2} + 2ab > {a^2} + {b^2} > {a^2} + {b^2} - 2ab > {a^2} - {b^2}\]
On substituting their values from equation \[\left( 2 \right)\], equation \[\left( 4 \right)\], equation \[\left( 6 \right)\] and equation \[\left( 8 \right)\], we get
\[{\left( {a - b} \right)^2} > {a^2} + {b^2} > {\left( {\left| a \right| - \left| b \right|} \right)^2} > \left| {{a^2} - {b^2}} \right|\]
Hence, we can easily see that the \[{\left( {a - b} \right)^2}\]has the greatest value.
Therefore, the correct option is option A.

Note: Here we have simplified the modulus algebraic expression. A modulus function is defined as a function which always gives the absolute value of a number or variable. It gives the magnitude of the number of variables. It is also called an absolute value function. This function always gives a positive value, no matter what input has been given to the function. That's the reason why we had taken the positive sign of \[b\] in equation \[\left( 2 \right)\].