Completing the Square Method for Solving Polynomial Equations

[crays]

hi guys.
If f(x) = x⁴ + 2x³ + 5x² - 16x - 20, show that f(x) can be expressed in the form (x² + x + a)² - 4(x + b)², where a and b are constant to be determined.

Hence, or otherwise, find both the real roots of the eqaution f(x) = 0. Find also the set of values of x such that f(x) > 0.

I tried completing the square
but i found
(x² + x + (4x+30)/8)² - 4[(x²)/16 + 4x + 185/32]

my a and b is WAY too off the answer, which is a = 4 and b = 3. Any help?

 

[Defennder]

You are given the answer in terms of a,b. So what you have to do is to expand out that answer and simply compare coefficients of powers of x to find out a,b.

 

[crays]

i have to find a and b first. I think i expand it wrongly. How should i complete the square for x^4 + 2x^3 + 5x^2 - 16x - 20 ? I tried again. I did :

(x^2 + x + 5/2)^2 - x^2 - 25/4 - 16x - 20. Correct?

 

[Defennder]

Did the question tell you to complete the square? If not, you have x^4 + 2x^3 + 5x^2 -16x - 20 = (x^2+x+a)^2 - 4(x+b)^2. All you need to do is to expand out the RHS and compare the coefficients to get the values of a,b.

 

[crays]

Oh ya... that is way easier ... dammit. Thanks.