If m-nx+28x^2+12x^3+9x^4 is a perfect square find the value of m and n
Answers
Answer:
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Answer:
m=16 and n =-16
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.