Alg1131W6T2 - Solving quadratics

Saturday, 14 April 2018

12:14 PM

Untitled picture.png Machine generated alternative text: The quadratic formula for solving az2 + bz + c = O is, of course, 2a Now that we know about complex numbers, we can interpret the square root in case the discriminant b2 — 4ac < O. In this case there is not a real square root, but there is a purely imaginary 2 2 square root ofthe form ± ri, since (pi) So for example, applying the quadratic formulato z + 8z + 41 Using complex numbers we can write the solution set as Notice that the ftvo zeros are complex conjugates both have the same real part Cl both have the same imaginary part Cl are negatives. 0 give the solutions: —8± 8 2 2 Note: the set {1 — i, 1 + i, 2} in Maple notation is written as {1+1 , 1-1 , 2} Untitled picture.png Machine generated alternative text: Use the quadratic formula to answer the following questions i) The rootsofz + 10 z + 106=0 are: _5_9i 2 ii) The roots of z • 1-5i -2Z+ 26 0 are:
Untitled picture.png Machine generated alternative text: Use the quadratic formula to answer the following questions i) The rootsofz + 10 z + 106=0 are: _5_9i 2 ii) The roots of z • 1-5i -2Z+ 26 0 are: Untitled picture.png Machine generated alternative text: re [here a: 5 01 tne equation o 48=2 + 676 = O and y = Untitled picture.png Machine generated alternative text: In the ordinary real plane, the line + y = 2 and the unit circle x + y 1 do not intersect, as you can see in the diagranm But if we extend our number system to the complex numbers, we can find two points [x, y] of intersection! They are given by [select all that apply] Untitled picture.png Machine generated alternative text: + 18.24
Untitled picture.png Machine generated alternative text: In the ordinary real plane, the line + y = 2 and the unit circle x + y 1 do not intersect, as you can see in the diagranm But if we extend our number system to the complex numbers, we can find two points [x, y] of intersection! They are given by [select all that apply] Untitled picture.png Machine generated alternative text: The polynomial p(z) z +1 factors into a product of the real linear factor Note: your factors should be monic, i.e. the highest power has coefficient 1 Hence the solutions to p(z) = O are [select all that apply] 1+iv'ä •v"ä+i _1+iy/9 O and the real quadratic factor
Untitled picture.png Machine generated alternative text: The polynomial p(z) z +1 factors into a product of the real linear factor Note: your factors should be monic, i.e. the highest power has coefficient 1 Hence the solutions to p(z) = O are [select all that apply] 1+iv'ä •v"ä+i _1+iy/9 O and the real quadratic factor

 

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