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 

From Complex Roots of a Quadratic Equation to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

  1. Worksheet on Triangle | Homework on Triangle | Different types|Answers

    Jun 21, 24 02:19 AM

    Find the Number of Triangles
    In the worksheet on triangle we will solve 12 different types of questions. 1. Take three non - collinear points L, M, N. Join LM, MN and NL. What figure do you get? Name: (a)The side opposite to ∠L…

    Read More

  2. Worksheet on Circle |Homework on Circle |Questions on Circle |Problems

    Jun 21, 24 01:59 AM

    Circle
    In worksheet on circle we will solve 10 different types of question in circle. 1. The following figure shows a circle with centre O and some line segments drawn in it. Classify the line segments as ra…

    Read More

  3. Circle Math | Parts of a Circle | Terms Related to the Circle | Symbol

    Jun 21, 24 01:30 AM

    Circle using a Compass
    In circle math the terms related to the circle are discussed here. A circle is such a closed curve whose every point is equidistant from a fixed point called its centre. The symbol of circle is O. We…

    Read More

  4. Circle | Interior and Exterior of a Circle | Radius|Problems on Circle

    Jun 21, 24 01:00 AM

    Semi-circular Region
    A circle is the set of all those point in a plane whose distance from a fixed point remains constant. The fixed point is called the centre of the circle and the constant distance is known

    Read More

  5. Quadrilateral Worksheet |Different Types of Questions in Quadrilateral

    Jun 19, 24 09:49 AM

    In math practice test on quadrilateral worksheet we will practice different types of questions in quadrilateral. Students can practice the questions of quadrilateral worksheet before the examinations

    Read More