find the values of 'h' and 'k' if 25x^4-20x^3+14x^2+hx+k is a perfect square.

Question

A' level maths

Answer 1

If 25x^4-20x^3+14x^2+hx+k is a perfect square, then the square root must be of the form (5x^2+ax+b), where a and b are constants.

(5x^2+ax+b)^2=25x^4+10ax^3+(a^2+10b)x^2+2abx+b^2. Comparing coefficients we see:

x^3: 10a=-20, so a=-2

x^2: a^2+10b=14, 4+10b=14, so b=1

x: 2ab=h=-4

constant: b^2=k=1