1)
i) use an algrbraic method to find the square root of 2+i√5. give your answer in the form x+iy, where x and y are exact real numbers
ii)hence find in the form x+iy where x and y are exact real numbers, the roots of the equation z^4-4z^2+9=0
iii) show those roots on an argand diagram (part ii)
iv) given that α is a root of the equation in part (ii) such that 0< or equal to argα< or equal to π/2, sketch on the same argand diagram the locus of |z-α|=|z|
2+i√5 = 1/4 (8+4i√5) = 1/4 (10+4i√5-2) = 1/4 (√10+i√2)^2
so √(2+i√5) = 1/2 (√10+√2i)
The rest should fall right out.
I don't understand how you got any of that =(
where'd the 1/4 and all that come from?
i) To find the square root of a complex number, we can use the algebraic method. Let's start with the given complex number 2 + i√5.
First, we express this number in the standard form a + bi, where a and b are real numbers.
2 + i√5 can be rewritten as 2 + √5i.
Now, we can find the square root of this complex number by applying the following steps:
1. Calculate the magnitude, or modulus, of the complex number: |z| = √(a^2 + b^2).
In this case, the magnitude of 2 + √5i is |z| = √(2^2 + (√5)^2) = √(4 + 5) = √9 = 3.
2. Calculate the argument, or angle, of the complex number: arg(z) = atan(b/a).
In this case, the argument of 2 + √5i is arg(z) = atan(√5/2).
3. Find the square root of the magnitude: √|z| = √3.
4. Divide the argument by 2: arg(z)/2 = atan(√5/4).
5. Apply the square root formula for complex numbers: √(√|z| * (cos(arg(z)/2) + i*sin(arg(z)/2))).
We substitute the values from steps 3 and 4: √3 * (cos(atan(√5/4)/2) + i*sin(atan(√5/4)/2)).
Therefore, the square root of 2 + i√5 is √3 * (cos(atan(√5/4)/2) + i*sin(atan(√5/4)/2)), which can be written as x + iy.
ii) Now let's solve the equation z^4 - 4z^2 + 9 = 0 using the square root found in part (i).
We can rewrite the equation as (z^2)^2 - 4z^2 + 9 = 0.
Let u = z^2. Substituting this into the equation, we have:
u^2 - 4u + 9 = 0
Using the quadratic formula, we can find the values of u:
u = (-(-4) ± √((-4)^2 - 4(1)(9))) / (2(1))
Simplifying, we get:
u = (4 ± √(16 -36)) / 2
u = (4 ± √(-20)) / 2
Since the discriminant is negative, it means u = z^2 does not have real solutions. However, since we found the complex square root in part (i), we can still find complex solutions.
Rewriting u using the square root: u = (4 ± 2i√5) / 2
Simplifying, we get:
u = 2 ± i√5
Now we solve for z:
z^2 = 2 ± i√5
Taking the square roots using the approach explained in part (i), we get:
z = ± (√3 * (cos(atan(√5/4)/2) + i*sin(atan(√5/4)/2)))
Hence, the roots of the equation z^4 - 4z^2 + 9 = 0 are ± (√3 * (cos(atan(√5/4)/2) + i*sin(atan(√5/4)/2))), where x and y are exact real numbers.
iii) To show these roots on an Argand diagram, plot the points corresponding to the real and imaginary parts of the roots. The real part corresponds to the x-coordinate, and the imaginary part corresponds to the y-coordinate.
iv) To sketch the locus of |z-α| = |z| on the same Argand diagram, we can use the following steps:
1. Plot the point α.
2. Plot several points on the Argand diagram that satisfy the equation |z-α| = |z|.
3. Connect all these points to form the locus.
Remember that |z| represents the distance of z from the origin, and |z-α| represents the distance of z from α. Thus, the locus represents all the points where the distances to both z and α are equal.