## If m-nx +28xsquare + 12xcube+9xpower4 is a perfect square, find the values of m and n

Question

If m-nx +28xsquare + 12xcube+9xpower4 is a perfect square, find the values of m and n

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1 month 2021-08-13T01:52:40+00:00 1 Answer 0 views 0

Step-by-step explanation:

It is given that the polynomial m−nx+28x

2

+12x

3

+9x

4

is a perfect square, we must equate it to the square of general form of equation that is (ax

2

+bx+c)

2

as shown below:

9x

4

+12x

3

+28x

2

−nx+m=(ax

2

+bx+c)

2

⇒9x

4

+12x

3

+28x

2

−nx+m=(ax

2

)

2

+(bx)

2

+(c)

2

+(2×ax

2

×bx)+(2×bx×c)+(2×c×ax

2

)

(∵(a+b+c)

2

=a

2

+b

2

+c

2

+2ab+2bc+2ca)

⇒9x

4

+12x

3

+28x

2

−nx+m=a

2

x

4

+b

2

x

2

+c

2

+2abx

3

+2bcx+2acx

2

Now, comparing the coefficients, we get:

a

2

=9,b

2

+2ac=28,c

2

=m,2ab=12,2bc=−n

a

2

=9

⇒a=3

2ab=12

⇒2×3×b=12

⇒6b=12

⇒b=2

b

2

+2ac=28

⇒2

2

+(2×3c)=28

⇒4+6c=28

⇒6c=28−4

⇒6c=24

⇒c=4

c

2

=m

⇒m=4

2

⇒m=16

2bc=−n

⇒n=−2bc

⇒n=−2×2×4

⇒n=−16

Hence, m=16 and n=−16.