x-2-x-1-3-5-7-9-11-13-x-2-2x-48-find-all-x-real-that-is-solution-of-above-equation-

Question Number 733 by 123456 last updated on 05/Mar/15

∣∣∣∣∣∣∣ x^{2} - x - 1 ∣ - 3 ∣ - 5 ∣ - 7 ∣ - 9 ∣ - 11 ∣ - 13 ∣

= x^{2} - 2 x - 48

f i n d a l l x r e a l t h a t i s s o l u t i o n o f a b o v e

e q u a t i o n

Answered by prakash jain last updated on 05/Mar/15

x^{2} - 2 x - 48 = (x - 8) (x + 6)

(x - 8) (x + 6) ⩾ 0 for x ⩾ 8 or x ⩽ - 6

Case I : x ⩾ 8

x^{2} - x - 1 > 0

\dots

x^{2} - x - 49 > 0

x^{2} - x - 49 = x^{2} - 2 x - 48

x = 1 \Rightarrow No solution for x ⩾ 8

Case II : x ⩽ - 6

x^{2} - x - 1 > 0

\dots

x^{2} - x - 36 > 0

x^{2} - x - 49 = (x - \frac{1 + \sqrt{197}}{2}) (x - \frac{1 - \sqrt{197}}{2})

x^{2} - x - 49 ⩾ 0 gives only solution x = 1 - 6

x^{2} - x - 49 < 0

- x^{2} + x + 49 = x^{2} - 2 x - 48

2 x^{2} - 3 x - 97 = 0

Root for x ⩽ - 6

x = \frac{3 - \sqrt{9 + 8 \times 97}}{4} = \frac{3 - \sqrt{9 + 776}}{4} = \frac{3 - \sqrt{785}}{4}

\frac{3 - \sqrt{785}}{4} - \frac{1 + \sqrt{197}}{2} = \frac{3 - \sqrt{785} - 2 - \sqrt{788}}{4} < 0

\frac{3 - \sqrt{785}}{4} - \frac{1 - \sqrt{197}}{2} = \frac{3 - \sqrt{785} - 2 + \sqrt{788}}{4} > 0

x^{2} - x - 49 < 0 for x = \frac{3 - \sqrt{785}}{4}

\frac{3 - \sqrt{785}}{4} + 6 = \frac{27 - \sqrt{785}}{4} = \frac{\sqrt{729} - \sqrt{785}}{4} < 0

\frac{3 - \sqrt{785}}{4} < - 6 and x^{2} - x - 49 < 0 for x = \frac{3 - \sqrt{785}}{4}

so it a valid solution .

Solution for real x

x = \frac{3 - \sqrt{785}}{4}