How do you solve the quadratic #7x^2+5=-137# using any method?
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There are no real solutions. But you can solve this using complex numbers. Then, the solutions (both imaginary) are:
#x_1 = + sqrt {142/7} cdot i and x_2 = - sqrt {142/7} cdot i#
You can probe there are no solutions in #RR# clearing the #x# as follows:
#7 x^2 + 5 = - 137 color(white) "." rArr color(white) "." 7 x^2 = - 137 - 5 = - 142#,
then
#7 x^2 = - 142 color(white) "." rArr color(white) "." x^2 = - 142/7 color(white) "." rArr color(white) "." x = +- sqrt {- 142/7} !in RR#.
However, if we still want to give an answer, we can use complex numbers using the definition of the imaginary number #i#:
Since #i = sqrt {- 1}# we can write the solutions of the quadratic equation of the form:
#x = +- sqrt {- 142/7} = +- sqrt {142/7} cdot sqrt {- 1} = +- sqrt {142/7} cdot i in CC#.
