Evaluate x²+19 / 2−x for x = 3i.

Question

Accepted Answer

Hello, everyone. We are asked to evaluate the given rational expression for X equals five. I, our rational expression is X squared plus 23 in parentheses, divided by the parentheses, seven minus X. We have four answer choices, all of which are rational numbers but also complex numbers. I'm gonna rewrite our initial expression. So X squared plus divided by seven minus X. And now I'm going to rewrite it with five I in for X. So five I squared plus divided by seven minus five. I squaring five I five squared is 25 I squared is I squared plus 23 still divided by seven minus five. I I recall I squared equals negative one. So 25 multiplied by negative one is negative plus 23 divided by seven minus five. I, I'm gonna combine the numbers in my numerator and then I notice that I have an imaginary number in my denominator and I don't want that to stay there. So when we combine our numerator, we have negative two divided by seven minus five I to get the I out of the denominator, we need to multiply top and bottom by its conjugate. So the same terms, but with a plus in between. So it's the opposite sign in between. So we had seven minus five I, we now are multiplying numerator and denominator by seven plus five I. And when we do so the numerator becomes negative minus 10. I our denominator, they are conjugates and they are also um the product of a sum and a difference. So we should end up with the difference of squares. So seven squared is 49 minus five I squared, which we did before is 25 I squared. Again, we're calling the I squared equals negative one. I'm going to rewrite this one more time. So I have 14 minus 10. I is my numerator. 49 is part of my denominator and then negative 25 multiplied by negative one would be plus 25. So now my denominator is 49 plus 25. My numerator is still negative 14 minus 10. I combining the numbers of my denominator. I get 74. So now I wanna see if I can reduce by anything. I notice everything I have. So I have negative 14 minus 10. I divided by 74. All of those terms can be reduced by two. So that would give me negative seven minus five, I divided by 37. And I don't think that can be reduced any further, but I wanna get it into standard form. So we're gonna separate this into two fractions. So we'll have negative seven 37th minus five I divided by 37. This is my answer and it matches answer choice. A have a nice day.

Struggling with Trigonometry?

Evaluate x²+19 / 2−x for x = 3i.

Watch next

Graphing Complex Numbers practice set