Can you please help me with problems 1d, 1e, 4 from the worksheet below? Thanks so much!

AT&T Department of Mathematics Bucknell University 9:45 PM Complex Numbers 32% Mathematics 202 Supplementary Handout We will now take a brief diversion to discuss & very important extension of Our idea of unumbrr". To motivate onr discusion we begin by trying to couple of quadratic equations. Equation A: Find the solutions Of z2 L z O. As tmial, start by trying to factor, but you quickly find you chn't,. so yog to the quadratic formula to get Equation B: Find the Solutions Of 22 = O. Here y.e get and we would conclude that our has no solutions because you can't take the square rc-xt of a negative number. But what if it did make to take the root Of a negative number? Specifically, denote the quantity by the symbol Now, in Che of Equation B, have Ir you play around With a few more examples, you Will quickly realia that if any quadratic equation with real coefficients hxs solutions that are not both then they have thc form r iy, where x and y real numbers. Definitions and terminology. A an of the form where and y are real numbers. Wc say that is the Of z, denotl Re(z), and V is the of z, denote (Note that ltn(z) is a uumber.) Complex numbers such that ROG) O (such i and iv'fi) are is n number, thon E is thc of Notice thne the of second quadratic equation in (l) are complex conjugates. As earlier. is true in general: solutions (having non-zem imaginary part) to quadratic equations with coefficients always in conjugate pairs. 1/6