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Section 3.1 #2: Express 21 in terms of i.
Solution: Compute 21=21i. Section 3.1 #12: Simplify. Write the answer in the form a+bi, where a and b are real numbers:(65i)+(9+2i).
Solution: Compute (65i)+(9+2i)=(6+9)+(5i+2i)=33i.

Section 3.1 #39: Simplify. Write the answer in the form a+bi, where a and b are real numbers: (1+3i)(14i). Solution: Compute (1+3i)(14i)=14i+3i12i2=1i+12=13i

Section 3.1 #74: Simplify. Write the answer in the form a+bi, where a and b are real numbers: 5+3i1i. Solution: Compute 5+3i1i=(5+3i1i)(1+i1+i)=5+5i+3i+3i21i2=5+(5+3)i31(1)=(53)+(5+3)i2=532+5+32i.

Section 3.1 #85: Simplify (i)71. Solution: We want to exploit the fact that i2=1. First rewrite this as (i)71=(1)71i71=(1)ii70. To finish, we compute (i)71=ii70=i(i2)35=i(1)35=i(1)=i.

Section 3.2 #34: Solve the quadratic equation x2+6x+13=0. Solution: Use the quadratic formula with a=1, b=6, and c=13 to compute x=6±624(1)(13)2=6±36522=6±162=6±4i2=3±2i.

Section 3.2 #42: Solve the quadratic equation 3t2+8t+3=0. Solution: Use the quadratic formula iwht a=3, b=8, and c=3 to compute x=8±824(3)(3)2(3)=8±64366=8±286=8±276=43±73.

Section 3.3 #4: Find the vertex, find the axis of symmetry, determine whether there is a maximum or a minimum value (and find it), and graph the function g(x)=x2+7x8. Solution: The coefficient of x2 is already 1 so to complete the square, add and subtract (72)2 and then factor to get x2+7x8=x2+7x8+(72)2(72)2=(x+72)28(72)2=(x+72)2324494=(x+72)2814. From this we see that the vertex is (72,814). The axis of symmetry is x=72. There is no maximum. The minimum value is 814 and it occurs as x=72.

Section 3.3 #15: Find the vertex, find the axis of symmetry, determine whether there is a maximum or a minimum value (and find it), and graph the function g(x)=2x2+2x+1. Solution: First factor out the 2 to get 2(x2x12). Inside, we will complete the square by adding and subtracting (12)2: 2(x2x12)=2(x2x12+(12)2(12)2)=2((x12)21214)=2((x12)234)=2(x12)2+64=2(x12)2+32. From this we see the vertex is (12,32). The axis of symmetry is x=12, there is no minimum, and the maximum is 32 which occurs at x=12.