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

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Answer 1 · 2018-05-16T12:01:52

If we consider a general quadratic equation:

$a x^{2} + b x + c = 0$ $ax^2 + bx+ c = 0$

And suppose that we denote roots by $α$ $alpha$ and $β$ $beta$ , then

$x = α, β \Rightarrow (x - α) (x - β) = 0$ $x=alpha, beta => (x-alpha)(x-beta) = 0$
$:. x^2 - (alpha+beta)x+alpha beta = 0$

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: $alpha, beta = 1+-2i$

$S= (1-2i) + (1+2i) = 2$
$P = (1-2i)(1+2i) = 1+4 = 5$

So the equation with these roots is:

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

Ex 2: $alpha, beta = 2+-1i$

$S= (2-i) + (2+i) = 4$
$P = (2-i)(2+i) = 4+1 = 5$

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: $alpha, beta = +-3i$

$S= (-3i) + (3i) = 0$
$P = (-3i)(3i) = 9$

So the equation with these roots is:

$x^2 - 0x+ 9= 0$
$:. x^2 + 9= 0$