Complex Roots of a Quadratic Equation

We will discuss about the complex roots of a quadratic equation.

In a quadratic equation with real coefficients has a complex root α + iβ then it has also the conjugate complex root α - iβ.

Proof:

To prove the above theorem let us consider the quadratic equation of the general form:

ax$^{2}$ + bx + c = 0 where, the coefficients a, b and c are real.

Let α + iβ (α, β are real and i = √-1) be a complex root of equation ax$^{2}$ + bx + c = 0. Then the equation ax$^{2}$ + bx + c = 0 must be satisfied by x = α + iβ.

Therefore,

a(α + iβ)$^{2}$ + b(α + iβ) + c = 0

or, a(α$^{2}$ - β$^{2}$ + i ∙ 2 αβ) + bα + ibβ + c = 0, (Since, i$^{2}$ = -1)

or, aα$^{2}$ - aβ$^{2}$ + 2iaαβ + bα + ibβ + c = 0,

or, aα$^{2}$ - aβ$^{2}$ + bα + c + i(2aαβ + bβ) = 0,

Therefore,

aα$^{2}$ - aβ$^{2}$ + bα + c = 0 and 2aαβ + bβ = 0

Since, p + iq = 0 (p, q are real and i = √-1) implies p = 0 and q = 0]

Now substitute x by α - iβ in ax$^{2}$ + bx + c we get,

a(α - iβ)$^{2}$ + b(α - iβ) + c

= a(α$^{2}$ - β$^{2}$ - i ∙ 2 αβ) + bα - ibβ + c, (Since, i$^{2}$ = -1)

= aα$^{2}$ - aβ$^{2}$ - 2iaαβ + bα - ibβ + c,

= aα$^{2}$ - aβ$^{2}$ + bα + c - i(2aαβ + bβ)

= 0 - i ∙ 0 [Since, aα$^{2}$ - aβ$^{2}$ + bα + c = 0 and 2aαβ + bβ = 0]

= 0

Now we clearly see that the equation ax$^{2}$ + bx + c = 0 is satisfied by x = (α - iβ) when (α + iβ) is a root of the equation. Therefore, (α - iβ) is the other complex root of the equation ax$^{2}$ + bx + c = 0.

Similarly, if (α - iβ) is a complex root of equation ax$^{2}$ + bx + c = 0 then we can easily proved that its other complex root is (α + iβ).

Thus, (α + iβ) and (α - iβ) are conjugate complex roots. Therefore, in a quadratic equation complex or imaginary roots occur in conjugate pairs.

Solved example to find the imaginary roots occur in conjugate pairs of a quadratic equation:

Find the quadratic equation with real coefficients which has 3 - 2i as a root (i = √-1).

Solution:

According to the problem, coefficients of the required quadratic equation are real and its one root is 3 - 2i. Hence, the other root of the required equation is 3 - 2i (Since, the complex roots always occur in pairs, so other root is 3 + 2i.

Now, the sum of the roots of the required equation = 3 - 2i + 3 + 2i = 6

And, product of the roots = (3 + 2i)(3 - 2i) = 3$^{2}$ - (2i)$^{2}$ = 9 - 4i$^{2}$ = 9 -4(-1) = 9 + 4 = 13

Hence, the equation is

x$^{2}$ - (Sum of the roots)x + product of the roots = 0

i.e., x$^{2}$ - 6x + 13 = 0

Therefore, the required equation is x$^{2}$ - 6x + 13 = 0.

11 and 12 Grade Math 

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