What method is required to find out "for

Sabrina Bonilla

Sabrina Bonilla

Answered question

2022-03-13

What method is required to find out "for what values of k is 4x2+kx+14 a perfect square?"

Answer & Explanation

molandco8fj

molandco8fj

Beginner2022-03-14Added 3 answers

According to this if b2=4ac then the expression is a perfect square.
So k2=4(4)(14)
So k=±2
See I am going to show you why we assume 4k2 as x2 and 4 as x.
So lets compare the 2 expressions-
a2x2+2abx+b2=4x2+kx+4
Now see if we consider 4x2 as b2 then notice this is wrong because 4x2 has x2 sign which means 4x2 can only considered with a term having x2. Thus we can say-
a2x2=4x2,2abx=kx,b2=4
Now you can compare and find the answer like you did in question
Porter Camacho

Porter Camacho

Beginner2022-03-15Added 6 answers

We consider a polynomial in x
p(x)=4x2+kx+14
which has degree 2, constant term 14, linear term kx and square term 4x2. We are looking for a representation as perfect square
q(x)=(ax+b)2
We recall two polynomials are equal, i.e. p=q if they have the same degree and the coefficients of like powers of x are equal.

p ( x ) = q ( x ) (1) 4 x 2 + k x + 1 4 = a 2 x 2 + 2 a b x + b 2

From (1) we see degp=degq and we obtain by comparison of coefficients
4=a2
k=2ab
14=b2

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