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S.O.S. Mathematics CyberBoard :: View topic - Where am I am making an error? #2
Z = a + bi
z^2 + 2iz + 3 = 0
a^2 + 2abi - b^2 + 2ai - 2b + 3 = 0
a^2 - b^2 - 2b+3 = 0
2ab + 2a = 0
b=-2a/2a = -1
but according to the answers it b should be 1 or -3.
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Plug in -1 for your b to get your a.
This gives a^2 = -4, or a - 2i or -2i,
which gives z = i or -3i, thus the actual a = 0 and b = 1 or -3.
OR
Use quadratic formula to solve for z.
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The jump from this line:
a^2 + 2abi - b^2 + 2ai - 2b + 3 = 0
to this line:
a^2 - b^2 - 2b+3 = 0
is wrong.
That second line should be:
a^2 - b^2 - 2b + 3 + 2ai(b+1) = 0
Try again from here
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Quote: : The jump from this line:
a^2 + 2abi - b^2 + 2ai - 2b + 3 = 0
to this line:
a^2 - b^2 - 2b+3 = 0
is wrong.
That second line should be:
a^2 - b^2 - 2b + 3 + 2ai(b+1) = 0
Try again from here
Actually, he's trying to equate real & imaginary parts:
a^2 + 2abi - b^2 + 2ai - 2b + 3 = 0 = 0 + 0i gives
a^2 - b^2 - 2b + 3 = 0
and
2abi + 2ai = 0i or 2ab + 2a = 0 or 2a(b + 1) = 0
What's a little confusing is this a turns out to be imaginary.
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[/Quote: ]
2abi + 2ai = 0i or 2ab + 2a = 0 or 2a(b + 1) = 0
What's a little confusing is this a turns out to be imaginary.[/Quote: ]
No, it doesn't have to.
Using your own work:
2a(b + 1) = 0
2a = 0
a = 0
Substitute this value back into
a^2 - b^2 - 2b + 3 = 0 and multiply (or divide)
both sides by a factor of -1 to get
b^2 + 2b - 3 = 0
(b - 1)(b + 3) = 0
b = 1, b = -3
So a = 0 and b = 1 =>
Z = 0 + i, or just i,
or the other solution
a = 0 and b = -3 =>
Z = 0 - 3i, or just -3i
Equation: z^2 + 2iz + 3 = 0
Check z = i:
(i)^2 + 2i(i) + 3 = 0
-1 + 2(-1) + 3 = 0
-1 - 2 + 3 = 0
0 = 0 It checks.
Check z = -3i:
(-3i)^2 + 2i(-3i) + 3 = 0
9i^2 - 6i^2 + 3 = 0
3i^2 + 3 = 0
3(-1) + 3 = 0
-3 + 3 = 0
0 = 0 It checks.
** alstat's post showing where a comes out as a plus or minus 2i
is ALSO true if you go down that route, but ultimately they will
produce the SAME set of solutions as I have given.
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Quote: : The jump from this line:
a^2 + 2abi - b^2 + 2ai - 2b + 3 = 0
to this line:
a^2 - b^2 - 2b+3 = 0
is wrong.
That second line should be:
a^2 - b^2 - 2b + 3 + 2ai(b+1) = 0
Try again from here
Well, my line can't really be incorrect?
I'm just comparing the imaginery and real values on both sides of the '=' separately?
edit: too late, forgot to F5
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Well, niggards solution is more simple to follow.
I don't get this though from the other solution...
This gives a^2 = -4, or a - 2i or -2i,
True, I see this.
"which gives z = i or -3i,"
How?
What am I missing?
"thus the actual a = 0 and b = 1 or -3."
??
Thanks.
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