Question

What should be added to each of the following to make it a perfect square?
I. \[4{x^2} + 24xy\]
II. \[9{x^2} + 48x\]
III. \[{x^2} + x\]
IV. \[{a^2} - 14ab\]
V. \[9{p^2} - 36pq\]
VI. \[121{a^2} + 154a\]

Answer 1

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. Also, we need to know the basic algebraic formulae to solve these types of questions. Also, we need to know the basic square and square root values for basic numbers to make an easy calculation.

Complete step by step solution:
Let’s solve the first expression which is given in the question.
I. \[4{x^2} + 24xy\]
We know that,
 \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[4{x^2} + 24xy\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( {2x} \right)^2} + \left( {2 \times 2x \times 6y} \right)\] (Here we would consider \[a = 2x\] and \[b = 6y\] )
Let’s add \[{\left( {6y} \right)^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( {2x} \right)^2} + \left( {2 \times 2x \times 6y} \right) + {\left( {6y} \right)^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[4{x^2} + 24xy\] is \[{\left( {2x} \right)^2} + \left( {2 \times 2x \times 6y} \right) + {\left( {6y} \right)^2}\] .

II. \[9{x^2} + 48x\]
We know that,
 \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[9{x^2} + 48x\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( {3x} \right)^2} + \left( {2 \times 3x \times 8} \right)\] (Here we would consider \[a = 3x\] and \[b = 8\] )
Let’s add \[{8^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( {3x} \right)^2} + \left( {2 \times 3x \times 8} \right) + {8^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[9{x^2} + 48x\] is \[{\left( {3x} \right)^2} + \left( {2 \times 3x \times 8} \right) + {8^2}\] .

III. \[{x^2} + x\]
We know that,
 \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[{x^2} + x\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( x \right)^2} + \left( {2 \times x \times \dfrac{1}{2}} \right)\] (Here we would consider \[a = x\] and \[b = \dfrac{1}{2}\] )
Let’s add \[{\left( {\dfrac{1}{2}} \right)^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( x \right)^2} + \left( {2 \times x \times \dfrac{1}{2}} \right) + {\left( {\dfrac{1}{2}} \right)^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[{x^2} + x\] is \[{\left( x \right)^2} + \left( {2 \times x \times \dfrac{1}{2}} \right) + {\left( {\dfrac{1}{2}} \right)^2}\] .

IV. \[{a^2} - 14ab\]
We know that,
 \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[{a^2} - 14ab\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( a \right)^2} - \left( {2 \times a \times 7b} \right)\] (Here we would consider \[a = a\] and \[b = 7b\] )
Let’s add \[{\left( {7b} \right)^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( a \right)^2} - \left( {2 \times a \times 7b} \right) + {\left( {7b} \right)^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[{a^2} - 14ab\] is \[{\left( a \right)^2} - \left( {2 \times a \times 7b} \right) + {\left( {7b} \right)^2}\] .

V. \[9{p^2} - 36pq\]
We know that,
 \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[9{p^2} - 36pq\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( {3p} \right)^2} - \left( {2 \times 3p \times 6q} \right)\] (Here we would consider \[a = 3p\] and \[b = 6q\] )
Let’s add \[{\left( {6q} \right)^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( {3p} \right)^2} - \left( {2 \times 3p \times 6q} \right) + {\left( {6q} \right)^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[9{p^2} - 36pq\] is \[{\left( {3p} \right)^2} - \left( {2 \times 3p \times 6q} \right) + {\left( {6q} \right)^2}\] .

VI. \[121{a^2} + 154a\]
We know that,
 \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\]
To find the perfect square of a given expression we have to convert the expression in the form of the RHS part of the above algebraic formula.
We have \[121{a^2} + 154a\]
So, the above equation can also be written as,
 \[ \Rightarrow {\left( {11a} \right)^2} + \left( {2 \times 11a \times 7} \right)\] (Here we would consider \[a = 11a\] and \[b = 7\] )
Let’s add \[{\left( 7 \right)^2}\] in the above equation
So, we get
 \[ \Rightarrow {\left( {11a} \right)^2} + \left( {2 \times 11a \times 7} \right) + {\left( 7 \right)^2}\] (It is in the form of the perfect square)
So, the final answer is,
The perfect square form \[121{a^2} + 154a\] is \[{\left( {11a} \right)^2} + \left( {2 \times 11a \times 7} \right) + {\left( 7 \right)^2}\]

Note: This question describes the arithmetic operations like addition/ subtraction/ multiplication/ division. To solve these types of questions we would convert the given expression in the form of the RHS part of basic algebraic formulae which is mentioned in the above calculation. The final answer would satisfy the algebraic formula for these types of questions.