Quiz #4915
❓ Complex Roots Challenge
10 Questions
Test your skills with complex numbers by solving an intriguing equation involving square roots. This quiz is perfect for students and enthusiasts of advanced algebra.
Score Progress
0 of 10 correct
0 of 10 answered
Question 1
What is a common first step when solving the equation √k + √-k = 15 in the complex plane?
A.
Assume k is a real number.
B.
Let √k = z.
C.
Square both sides immediately.
D.
Substitute k = x + yi.
Question 2
If √k = z, what is √-k in terms of z, assuming principal roots and that √(-1) = i?
A.
z/i
B.
-z
C.
iz
D.
-iz
Question 3
After substituting √k = z and √-k = iz, what is the resulting linear equation for z?
A.
z^2 - z = 15i
B.
z + iz = 15
C.
z - iz = 15
D.
z^2 + (iz)^2 = 15
Question 4
From the equation z(1 + i) = 15, what is the first step to isolate z?
A.
Subtract i from both sides.
B.
Divide both sides by (1 + i).
C.
Multiply by (1 - i).
D.
Divide both sides by 15.
Question 5
To simplify z = 15 / (1 + i), which expression should you multiply the numerator and denominator by?
A.
(1 + i)
B.
i
C.
(1 - i)
D.
-i
Question 6
What is the value of z in the equation z = 15 / (1 + i)?
A.
-7.5 - 7.5i
B.
7.5 + 7.5i
C.
15 - 15i
D.
7.5 - 7.5i
Question 7
If z = √k, and you've found z, how do you calculate k?
A.
k = z
B.
k = √z
C.
k = z^2
D.
k = 1/z
Question 8
If z = a + bi, what is z^2?
A.
a^2 + b^2
B.
a^2 - b^2 + 2abi
C.
a^2 - b^2
D.
a^2 + b^2 - 2abi
Question 9
Given z = 7.5 - 7.5i, what is the value of k = z^2?
A.
-112.5i
B.
112.5
C.
112.5i
D.
-112.5
Question 10
If the problem specified that 'k' must be a real number, how many solutions would there be for √k + √-k = 15?
A.
Exactly one solution.
B.
Infinitely many solutions.
C.
Exactly two solutions.
D.
No real solutions.
Prompt: √k +√-k=15 finf k
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