Roots of a Quartic

Algebra Level 3

$\large f(x)^2+3f(x)+2$

If $f(x)= x^2-8x$ , then the equation above has 4 distinct real roots of the form $4 \pm{\sqrt{a}}$ for real value $a$ .

Denote the sum of all values of $a$ as $S$ . Evaluate $\dfrac {1740}S$ .

3 solutions

Devin Ky
Jun 24, 2015

Vishwak Srinivasan
Jun 26, 2015

$f(x)^2 + 3f(x) + 2 = (f(x) + 1)(f(x) + 2)$

$\Rightarrow f(x)^2 + 3f(x) + 2 = (x^2 - 8x + 1)(x^2 - 8x + 2) = 0$

$x^2 - 8x + 1 = 0 \Rightarrow x = 4 \pm \sqrt{15}$

$x^2 - 8x + 2 = 0 \Rightarrow x = 4 \pm \sqrt{14}$

Let the values of $a$ be $a_1 , a_2$

$S = a_1 + a_2 = 29 \Rightarrow \dfrac{1740}{S} = \dfrac{1740}{29} = \boxed{60}$

Mehul Bafna
Jun 20, 2015

f(x) = -1 or f(x)=-2 so solving it we get either 4+√15 or 4 - √15 and 4+√14 or 4-√14 So S = 15+14 =29. 1740/29 =60