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A quadratic equation has the form:

x^2 - (sum of the roots)x + (product of the roots) = 0

If the roots are imaginary roots, these roots are complex number a+bi and its conjugate a - bi, where a is the real part and b is the imaginary part of the complex number.

Their sum is:

a + bi + a - bi = 2a

Their product is:

(a + bi)(a - bi) = a^2 + b^2

Thus the equation will be in the form:

x^2 - 2a(x) + a^2 + b^2 = 0 or,

x^2 - 2(real part)x + [(real part)^2 + (imaginary part)^2]= 0

For example if the roots are 3 + 5i and 3 - 5i, the equation will be:

x^2 - 2(3)x + 3^2 + 5^2 = 0

x^2 - 6x + 34 = 0 where,

a = 1, b = -6, and c = 34.

Look at the denominator of this quadratic equation:

D = b^2 - 4ac.

D = (-6)^2 - (4)(1)(34) = 36 - 136 = -100

D < 0

Since D < 0 this equation has two imaginary roots.

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Q: What are examples of quadratic formula with 2 imaginary roots?
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