How do you find the value of the discriminant and determine the nature of the roots #x^2 + 4x = -7#?

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Answer 1 · 2016-09-26T09:31:28

Roots are complex conjugates. For given equation they are #x=-2-3i# or #x=-2+3i#

The discriminant of a quadratic equation #ax^2+bx+c=0# is #b^2-4ac#, which decides the nature of roots of the equation.

If #a#, #b# and #c# are rational and #b^2-4ac# is square of a rational number, roots are rational.

If #b^2-4ac>0# but is not a square of a rational number, roots are real but not rational.

If #b^2-4ac>0-0# we have equal roots.

If #b^2-4ac<0# roots are complex, and if #a#, #b# and #c# are rational. they are complex conjugates

In #x^2+4x=-7hArrx^2+4x+7=0#

the discriminant is #4^2-4xx1xx7=16-28=-12#

hence roots are complex conjugates.

In fact #x^2+4x+7=0#

#hArrx^2+4x+4-(-3)=0#

or #(x+2)^2-(3i^2)=0#

or #(x+2+3i)(x+2-3i)=0#

i.e. #x=-2-3i# or #x=-2+3i#