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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:(−6−5i)+(9+2i).
Solution: Compute
(−6−5i)+(9+2i)=(−6+9)+(−5i+2i)=3−3i.
Section 3.1 #39: Simplify. Write the answer in the form a+bi, where a and b are real numbers:
(1+3i)(1−4i).
Solution: Compute
(1+3i)(1−4i)=1−4i+3i−12i2=1−i+12=13−i
Section 3.1 #74: Simplify. Write the answer in the form a+bi, where a and b are real numbers:
√5+3i1−i.
Solution: Compute
√5+3i1−i=(√5+3i1−i)(1+i1+i)=√5+√5i+3i+3i21−i2=√5+(√5+3)i−31−(−1)=(√5−3)+(√5+3)i2=√5−32+√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±√62−4(1)(13)2=−6±√36−522=−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±√82−4(3)(3)2(3)=−8±√64−366=−8±√286=−8±2√76=−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+7x−8.
Solution: The coefficient of x2 is already 1 so to complete the square, add and subtract (72)2 and then factor to get
x2+7x−8=x2+7x−8+(72)2−(72)2=(x+72)2−8−(72)2=(x+72)2−324−494=(x+72)2−814.
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(x2−x−12).
Inside, we will complete the square by adding and subtracting (−12)2:
−2(x2−x−12)=−2(x2−x−12+(−12)2−(−12)2)=−2((x−12)2−12−14)=−2((x−12)2−34)=−2(x−12)2+64=−2(x−12)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.