Answer:
[tex] \purple{ \boxed{\bold{x = \frac{3 (\sqrt{5} - 1) }{2}}}} \: or \: \red{ \boxed{\bold{x = \frac{ - 3 (\sqrt{5} + 1) }{2}}}}[/tex]
Step-by-step explanation:
[tex] {x}^{2} + 3x - 9 = 0 \\ {x}^{2} + 2. \frac{3}{2}x = 9 \\ {x}^{2} + 2. \frac{3}{2} x + { \bigg (\frac{3}{2} \bigg)}^{2} = 9 + { \bigg (\frac{3}{2} \bigg)}^{2} \\ \\ \bigg(x + \frac{3}{2} \bigg)^{2} = 9 + \frac{9}{4} \\ \\ \bigg(x + \frac{3}{2} \bigg)^{2} = \frac{45}{4} \\ \\\bigg(x + \frac{3}{2} \bigg) = \pm \sqrt{ \frac{45}{4} } \\ \\\bigg(x + \frac{3}{2} \bigg) = \pm { \frac{3 \sqrt{5} }{2} } \\ \\ x = \pm { \frac{3 \sqrt{5} }{2} } - \frac{3}{2} \\ \\ x = \pm { \frac{3 \sqrt{5} - 3 }{2} } \\ \\ x = \frac{3 \sqrt{5} - 3 }{2} \: or \: x = \frac{ - 3 \sqrt{5} - 3 }{2} \\ \\ \purple{ \boxed{\bold{x = \frac{3 (\sqrt{5} - 1) }{2}}}} \: or \: \red{ \boxed{\bold{x = \frac{ - 3 (\sqrt{5} + 1) }{2}}}} \\ [/tex]
