Find-the-sum-of-all-the-real-number-x-that-satisfy-2x-2-5x-1-2x-3-1-

Question Number 144260 by ZiYangLee last updated on 23/Jun/21

Find the sum of all the real number

x that satisfy {(2 x^{2} + 5 x + 1)}^{2 x - 3} = 1

Answered by Ar Brandon last updated on 24/Jun/21

2 x - 3 = 0 \lor {2 x}^{2} + 5 x + 1 = 1

x = \frac{3}{2} \lor x = 0, x = - \frac{5}{2}

{2 x}^{2} + 5 x + 1 > 0

Answered by ArielVyny last updated on 24/Jun/21

\boldsymbol s o l u t i o n 1 {(1)}^{α} = 1

2 x^{2} + 5 x + 1 = 1 x (2 x + 5) = 0

x = 0 x = - \frac{5}{2}

You can't use 'macro parameter character #' in math mode

2 x - 3 = 0 x = \frac{3}{2}

\boldsymbol s o l u t i o n 3 {(- 1)}^{2 \boldsymbol n} = 1

2 x^{2} + 5 x + 1 = - 1

2 x^{2} + 5 x + 2 = 0

2 [{(x + \frac{5}{2})}^{2} - \frac{25}{4} + \frac{8}{4}] = 0

2 [{(x + \frac{5}{2})}^{2} - \frac{17}{4}] = 0

2 [(x + \frac{5}{2} - \frac{\sqrt{17}}{2}) (x + \frac{5}{2} + \frac{\sqrt{17}}{2})] = 0

x_{1} = - \frac{5}{2} + \frac{\sqrt{17}}{2} a n d x_{2} = - \frac{5}{2} - \frac{\sqrt{17}}{2}

t h e n x_{1} a n d x_{2} a r e n o t p e e r

s o i n c a s e 3 t h e r e i s n o t s o l u t i o n

S = {0 . \frac{3}{2} . - \frac{5}{2}}

w e h a v e 0 + \frac{3}{2} - \frac{5}{2} = - 1