Question

Find the values of a and b if 16x^4 - 24x^3 +(a-1)x^2 +(b+1)x+49 is a perfect square.
Please explain​

Answer 1

The values of a is 66 and b is -43

Step-by-step explanation:

Given polynomial

$p(x)=16x^4-24x^3+(a-1)x^2+(b+1)x+49$

The perfect square polynomial will have 3 terms first term and last term is perfect square and the middle term which must have the power of x as 1 must also be perfect square.

Using $(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

$p(x)=(4x^2)^2+7^2+(3x)^2+2\times(4x^2)\times (-3x)+2\times 7\times 4x^2+2\times 7\times (-3x)$

$\implies p(x)=16x^4+49+9x^2-24x^3+56x^2-42x$

$\implies p(x)=16x^4-24x^3+65x^2-42x+49$

On comparing we get

$a-1=65$

$\implies a=66$

And

$b+1=-42$

$\implies b=-43$

The perfect square is

$p(x)=(2x^2-3x+7)^2$

Note: There can be one more case in which we take the term $4x^2$ as a negative term.

Hope this answer is helpful.

Know More:

Q: For what value of k, (4-k)x2 + (2k + 4)x + (8k + 1) = 0, is a perfect square?

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