First, put the equation in standard quadratic form:
#(x - 3)^2 = 5#
#(x - 3)(x - 3) = 5#
#x^2 - 3x - 3x + 9 = 5#
#x^2 - 6x + 9 = 5#
#x^2 - 6x + 9 - color(red)(5) = 5 - color(red)(5)#
#x^2 - 6x + 4 = 0#
We can now use the quadratic equation to solve this problem:
The quadratic formula states:
For #color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0#, the values of #x# which are the solutions to the equation are given by:
#x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))#
Substituting:
#color(red)(1)# for #color(red)(a)#
#color(blue)(-6)# for #color(blue)(b)#
#color(green)(4)# for #color(green)(c)# gives:
#x = (-color(blue)(-6) +- sqrt(color(blue)(-6)^2 - (4 * color(red)(1) * color(green)(4))))/(2 * color(red)(1))#
#x = (6 +- sqrt(36 - 16))/2#
#x = (6 +- sqrt(20))/2#
#x = (6 - sqrt(4 * 5))/2#; #x = (6 + sqrt(4 * 5))/2#
#x = (6 - sqrt(4)sqrt(5))/2#; #x = (6 + sqrt(4)sqrt(5))/2#
#x = (6 - 2sqrt(5))/2#; #x = (6 + 2sqrt(5))/2#
#x = 6/2 - (2sqrt(5))/2#; #x = 6/2 + (2sqrt(5))/2#
#x = 3 - sqrt(5)#; #x = 3 + sqrt(5)#
