We can 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)(-10)# for #color(blue)(b)#
#color(green)(41)# for #color(green)(c)# gives:
#x = (-color(blue)(-10) +- sqrt(color(blue)(-10)^2 - (4 * color(red)(1) * color(green)(41))))/(2 * color(red)(1))#
#x = (10 +- sqrt(100 - 164))/2#
#x = (10 +- sqrt(-64))/2#
#x = (10 +- sqrt(64 * -1))/2#
#x = (10 +- sqrt(64) * sqrt(-1))/2#
#x = (10 +- 8sqrt(-1))/2#
#x = 10/2 +- 8sqrt(-1)/2#
#x = 5 +- 4sqrt(-1)#
#x = 5 - 4sqrt(-1); x = 5 + 4sqrt(-1)#
In mathematics #sqrt(-1)# is often times represented as #i# representing an imaginary number. Using this we can rewrite the solution as:
#x = 5 - 4i; x = 5 + 4i#
