ANSWER
[tex]x = \frac{ 5 }{8} + \frac{\sqrt{ 103}i}{8} [/tex]
or
[tex]x = \frac{ 5 }{8} - \frac{\sqrt{ 103}i}{8} [/tex]
EXPLANATION
The given equation is:
[tex]4 {x}^{2} - 3x + 9 = 2x + 1[/tex]
We rewrite in the form ax² +bx+c=0 to get:
[tex]4 {x}^{2} - 3x - 2x + 9 - 1= 0[/tex]
[tex]4 {x}^{2} - 5x + 8= 0[/tex]
This implies that, a=4, b=-5,c=8.
The quadratic formula is given by,
[tex]x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} [/tex]
We plug in the values to get,
[tex]x = \frac{ - - 5\pm \sqrt{ {( - 5)}^{2} - 4(4)(8)} }{2(4)} [/tex]
[tex]x = \frac{ 5\pm \sqrt{ - 103} }{8} [/tex]
[tex]x = \frac{ 5\pm \sqrt{ 103} i}{8} [/tex]
[tex]x = \frac{ 5 }{8} + \frac{\sqrt{ 103}i}{8} [/tex]
Or
[tex]x = \frac{ 5 }{8} - \frac{\sqrt{ 103}i}{8} [/tex]
