
The value of n for which the expression $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ becomes a perfect square is
(A) 12
(B) 16
(C) 18
(D) 24
Answer
494.4k+ views
Hint:To solve this type of particular problem we use algebraic expression of square term.Let assume $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ = ${\left( {a{x^2} + bx + c} \right)^2}$.Expand the R.H.S term i.e assumed value by using formula ${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2zx$.After substituting assumed value in formula we have to compare the similar terms to get our answer.
Complete step-by-step answer:
Let assume $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ = ${\left( {a{x^2} + bx + c} \right)^2}$
We assume this because the question says $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ is a perfect square.
Now we expand ${\left( {a{x^2} + bx + c} \right)^2}$by using formula ${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2zx$
So ${\left( {a{x^2} + bx + c} \right)^2} = {a^2}{x^4} + {b^2}{x^2} + {c^2} + 2ab{x^3} + 2bcx + 2ac{x^2}$
Now ${a^2}{x^4} + {b^2}{x^2} + {c^2} + 2ab{x^3} + 2bcx + 2ac{x^2}$=$9{x^4} - 12{x^3} + n{x^2} - 8x + 4$
So now we compare terms
Firstly comparing ${x^4}$ we get ${a^2} = 9$
And $a = \pm 3$
Now compare constant term ${c^2} = 4$
And $c = \pm 2$
Now comparing ${x^3}$ term we get
$2ab = - 12$
And $ab = - 6$
Now we have to put value of $a$ so that we get
$b = \pm 2$
Now comparing ${x^2}$ we get
$2ac + {b^2} = n$
Now we have to put the value of a,b,c in the above equation to find the value of n.
$2 \times 3 \times 2 + {\left( { \pm 2} \right)^2}=n$
So the square of any number is always positive
$12 + 4$
$n = 16$
The value of $n = 16$
So, the correct answer is “Option B”.
Note:In this type of question we have to remember all the algebraic expressions and formulas that how we can use them .We have to compare all the similar terms and remember whenever we solve a square term we have to always use \[ \pm \] both signs so that we get all possible values that we need.
Complete step-by-step answer:
Let assume $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ = ${\left( {a{x^2} + bx + c} \right)^2}$
We assume this because the question says $9{x^4} - 12{x^3} + n{x^2} - 8x + 4$ is a perfect square.
Now we expand ${\left( {a{x^2} + bx + c} \right)^2}$by using formula ${\left( {x + y + z} \right)^2} = {x^2} + {y^2} + {z^2} + 2xy + 2yz + 2zx$
So ${\left( {a{x^2} + bx + c} \right)^2} = {a^2}{x^4} + {b^2}{x^2} + {c^2} + 2ab{x^3} + 2bcx + 2ac{x^2}$
Now ${a^2}{x^4} + {b^2}{x^2} + {c^2} + 2ab{x^3} + 2bcx + 2ac{x^2}$=$9{x^4} - 12{x^3} + n{x^2} - 8x + 4$
So now we compare terms
Firstly comparing ${x^4}$ we get ${a^2} = 9$
And $a = \pm 3$
Now compare constant term ${c^2} = 4$
And $c = \pm 2$
Now comparing ${x^3}$ term we get
$2ab = - 12$
And $ab = - 6$
Now we have to put value of $a$ so that we get
$b = \pm 2$
Now comparing ${x^2}$ we get
$2ac + {b^2} = n$
Now we have to put the value of a,b,c in the above equation to find the value of n.
$2 \times 3 \times 2 + {\left( { \pm 2} \right)^2}=n$
So the square of any number is always positive
$12 + 4$
$n = 16$
The value of $n = 16$
So, the correct answer is “Option B”.
Note:In this type of question we have to remember all the algebraic expressions and formulas that how we can use them .We have to compare all the similar terms and remember whenever we solve a square term we have to always use \[ \pm \] both signs so that we get all possible values that we need.
Recently Updated Pages
The correct geometry and hybridization for XeF4 are class 11 chemistry CBSE

Water softening by Clarks process uses ACalcium bicarbonate class 11 chemistry CBSE

With reference to graphite and diamond which of the class 11 chemistry CBSE

A certain household has consumed 250 units of energy class 11 physics CBSE

The lightest metal known is A beryllium B lithium C class 11 chemistry CBSE

What is the formula mass of the iodine molecule class 11 chemistry CBSE

Trending doubts
Worlds largest producer of jute is aBangladesh bIndia class 9 social science CBSE

Distinguish between Conventional and nonconventional class 9 social science CBSE

Draw an outline map of India and mark the following class 9 social science CBSE

soil is formed by the weathering of basalt rocks A class 9 social science CBSE

Write a short note on The Shiwalik Range class 9 social science CBSE

What is chronic hunger and seasonal hunger
