SOLUTION: Determine the quadratic equation whose roots have a sum of 12 and the roots difference is 4i. (Write a let statement to identify your roots)

Question 253132: Determine the quadratic equation whose roots have a sum of 12 and the roots difference is 4i. (Write a let statement to identify your roots)
Found 3 solutions by richwmiller, Alan3354, MathTherapy:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!

Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
a and b are the roots.
a + b = 12
a - b = 4i
----------
2b = 12 + 4i
b = 6 + 2i
a = 6 - 2i
-----------
f(x) = (x - 6 + 2i)*(x - 6 - 2i)
= x^2 - 12x + 40
----------------

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

The discriminant -16 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

In the field of imaginary numbers, the square root of -16 is + or - .

The solution is , or
Here's your graph:

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
To find any quadratic equation of the form , we have to realize that:

a = 1 (always)
b = - (sum of roots)
c = product of roots

Since a is always 1, and the sum of the roots = 12, then b = - (12) = -12

We now have:

To find c, we need to 1st determine the roots and multiply them

Let root 1 be , and root 2,

Since the sum of the roots = 12, then ----- --------- eq (i)
Also, since the roots� difference = 4i, then --------- eq (ii)

Adding equations (i) & (ii), we get:

or

Substituting 6 + 2i for in eq (i), we get:

Since we now have both roots, and , we multiply these two roots to get �c.�

Therefore, �c� = (6 + 2i)(6 � 2i) = , or, 36 � 4(-1) = 40
With �a� being 1, �b� being � 12, and �c� being 40, the quadratic equation in the form =
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