If we use the quadratic formula to factor #color(red)1x^2color(blue)(-3)xcolor(green)(-32)#
we find zeros at
#color(white)("XXX")x=(-(color(blue)(-3))+-sqrt((color(blue)(-3))^2-4 * color(red)1 * (color(green)(-32))))/(2 * color(red)1)#
#color(white)("XXXX")=(3+-sqrt(137))/2#
giving the factors:
#color(white)("XXX")x^2-3x-32 =(x-(3+sqrt(137))/2) * (x - (3-sqrt(137))/2)#
#x^2-3x-32# will be greater than zero if and only if
#{:
("both terms are positive","or","both terms are negative"),
(x>(3+sqrt(137))/2,,x < (3-sqrt(137))/2)
:}#
If you like you could use a calculator to evaluate these irrational expressions:
#x > 7.352349955color(white)("XX")orcolor(white)("XX")x < -4.35234995#
We could verify that these results are reasonable by examining the graph of #x^2-3x-32# and noting the values of #x# for which the graph (#y#) is above the X-axis:
