The sum of the two roots of x^4 - 8x^3 + 19x^2 + 4Ax + 2 = 0 is equal to the sum of other roots. Find A and solve the equation.
Answers
Answer:
A = -3
The four roots are 2±√2 and 2±√3.
Step-by-step explanation:
Call the four roots a, b, c and d, where a+b = c+d.
By the coefficient of x³, we know that a+b+c+d = 8, so a+b = c+d = 4.
By the coefficient of x², we know that
ab + ac + ad + bc + bd + cd = 19
⇒ ab + cd + (a+b)(c+d) = 19
⇒ ab + cd + 16 = 19
⇒ ab + cd = 3 ... (*)
The coefficient of x is then
-4A = abc + bcd + cda + dab = ab(c+d) + cd(a+b) = 4ab + 4cd
⇒ A = - (ab + cd) = -3
Continuing to find the roots, the constant coefficient tells us that
abcd = 2
Together with (*) above, it follows that ab and cd are the roots of the quadratic
x² - 3x + 2 = ( x - 1 ) ( x - 2 )
So we take ab=1 and cd=2.
Together with a+b = c+d = 4, we have
(x-a)(x-b) = x² - 4x + 1 ⇒ a and b are 2±√3
and
(x-c)(x-d) = x² - 4x + 2 ⇒ c and d are 2±√2
Hope this helps!