# Find the values and roots of the equation #z^4-2z^3+7z^2-4z+10=0#?

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Find the values of #a in RR# for which #ai# is a solution of

#z^4-2z^3+7z^2-4z+10=0#

Also find all the roots of this equation.

Find the values of

Also find all the roots of this equation.

##### 2 Answers

The roots are:

#### Explanation:

Given:

#z^4-2z^3+7z^2-4z+10 = 0#

If

#0 = (ai)^4-2(ai)^3+7(ai)^2-4(ai)+10#

#color(white)(0) = (a^4-7a^2+10)+2a(a^2-2)i#

#color(white)(0) = (a^2-5)(a^2-2)+2a(a^2-2)i#

#color(white)(0) = ((a^2-5)+2ai)(a^2-2)#

Hence

So two of the roots of the original quartic are

#(z-sqrt(2)i)(z+sqrt(2)i) = z^2+2#

We find:

#z^4-2x^3+7z^2-4z+10 = (z^2+2)(z^2-2z+5)#

#color(white)(z^4-2x^3+7z^2-4z+10) = (z^2+2)(z^2-2z+1+4)#

#color(white)(z^4-2x^3+7z^2-4z+10) = (z^2+2)((z-1)^2-(2i)^2)#

#color(white)(z^4-2x^3+7z^2-4z+10) = (z^2+2)(z-1-2i)(z-1+2i)#

So the other two roots are:

#z = 1+-2i#

#### Explanation:

We can use the fact that the problem asks us to find values of

Let us denote the other two solutions as

We can expand this, group like terms, and find that:

Set like-coefficients equal (so set

By graphing, we see that this solution set works.