# What is an example of a quadratic equation with imaginary roots?

##### 1 Answer

If we consider a general quadratic equation:

# ax^2 + bx+ c = 0#

And suppose that we denote roots by

# x=alpha, beta => (x-alpha)(x-beta) = 0#

# :. x^2 - (alpha+beta)x+alpha beta = 0#

Equivalently we can write as

# :. x^2 - ("sum of roots")x+("product of roots") = 0#

And comparing these identical equations we can readily derive the following important relationships:

# "sum of roots" = -b/a# and# "product of roots" = c/a#

We also know that complex roots appear in conjugate pairs, so we can form some suitable equations.

**Ex 1:**

# S= (1-2i) + (1+2i) = 2 #

# P = (1-2i)(1+2i) = 1+4 = 5 #

So the equation with these roots is:

# x^2 - 2x+5 = 0#

**Ex 2:**

# S= (2-i) + (2+i) = 4 #

# P = (2-i)(2+i) = 4+1 = 5 #

So the equation with these roots is:

# x^2 - 4x+ 5= 0#

If we strictly answer the question and require imaginary roots then we have no real component so:

**Ex 3:**

# S= (-3i) + (3i) = 0 #

# P = (-3i)(3i) = 9 #

So the equation with these roots is:

# x^2 - 0x+ 9= 0#

# :. x^2 + 9= 0#