Question 164565
"Zeros" or "roots" are simply math speak for values of x that make the ENTIRE equation equal to zero (note: "zeros" are not necessarily equal to zero)



So this means that we set the right side equal to zero like this:


{{{(x-7)^2(x^2+7)=0}}}



{{{(x-7)^2=0}}} or {{{x^2+7=0}}} Use the zero product property to break up the factors



Let's solve the first equation: {{{(x-7)^2=0}}} 



{{{(x-7)^2=0}}} Start with the given equation



{{{x-7=0}}} Take the square root of both sides to eliminate the square on the left side.



{{{x=7}}} Add 7 to both sides.



So the first solution is {{{x=7}}}



----------------------------------



Now let's solve the second equation: {{{x^2+7=0}}} 



{{{x^2+7=0}}} Start with the given equation



{{{x^2=-7}}} Subtract 7 from both sides.



{{{x=sqrt(-7)}}} Take the square root of both sides.



Since you CANNOT take the square root of a negative number, this means:


a) there are NO real solutions (for this part), and 

b) there are two complex (ie imaginary) solutions (if you have never heard of complex/imaginary solutions, then just ignore this next part)


So the two complex imaginary solutions are {{{x=i*sqrt(7)}}} or {{{x=-i*sqrt(7)}}}



===========================================


Answer:



So the solution(s) are


{{{x=7}}}, {{{x=i*sqrt(7)}}} or {{{x=-i*sqrt(7)}}}



Ignore the last two solutions if you have never heard of complex/imaginary solutions before.