Question

Find the values of m and n if x^4- 8x^3+(m +16)x^2+(n-16)x +16 is a perfect square.​

Answer 1

Answer:

m = 8 and n = -16

Step-by-step explanation:

let assume that

$(ax^{2} +bx^{2} +c)^2$  is a perfect square,

which means it follows the following conditions such as

let

$(x^4-8x^3 +(m+16)x^2+(n-16)x+16)= (ax^2+ bx+c)^2\\x^4-8x^3 +(m+16)x^2+(n-16)x+16= a^2x^4+b^2x^2+c^2+2abx^3+2bcx+2acx^2$

now, compare the respective coefficient of x to find the values of m and n

compare the coefficient of $x^4$,

$a^2=1\\a=1$

compare the coefficient of $x^3$

$2ab=-8\\2(1)b=-8\\b=\frac{-8}{2} \\b=-4$

also we get,

$c^2=16\\\\c=4$

now compare the coefficient of $x^2$ and $x$

$2ac+b^2=m+16\\2(1)(4)+(-4)^2=m+16\\\\8+16=m+16\\m=8$

$n-16=2bc\\n-16=2(-4)(4)\\n-16=-32\\n=-32+16\\n=-16$

thus above equation to be perfect square then value of m=8 and n= -16