How to Find the Roots of a Quadratic Equation

An equation of the form ax+ bx + c = 0 is referred to as a quadratic equation. Here ‘a’ should not be equal to zero. a, b, and c are real numbers. It has many applications in real-time situations.

The values of x which satisfy the quadratic equation are known as the roots of the quadratic equation. This equation will always have two roots. The roots will be either real or imaginary. In this article, we will discuss how to find the roots of a quadratic equation.

Formula to find the Roots of a Quadratic Equation?

We use the quadratic formula to find the roots of a quadratic equation.

The formula is given as

b±b24ac2a\frac{-b \pm \sqrt{b2-4ac}}{2a}

Here b– 4ac is called the discriminant. It is denoted by D.

Nature of Roots

Depending on the value of D, the nature of roots will change.

1. If D = 0, then the roots will be equal and real.

2. If D > 0, then the roots will be real and distinct.

3. If D < 0, then the roots will be imaginary.

If α and β are the roots of the equation, then

Sum of roots = -b/a

Product of roots = c/a.

Example

If α and β are the roots of the equation x– x + 1 = 0, then α2009 + β2009 =

(1) -1

(2) 1

(3) 2

(4) -2

Solution:

Given equation is x2-x+1 = 0

We find roots using quadratic formula.

x = 1±(14)2\frac{ -1 \pm \sqrt{(1-4)}}{2}

α = (-1+√3i)/2

β = (-1-√3i)/2

α = cos (π/3) +isin (π/3)

β = cos (π/3) – isin (π/3)

α2009 + β2009 = 2cos 2009(π/3)

= 2cos (668π+π+2π/3)

= 2cos(π+2π/)

= -2cos(2π/3)

= -2(-½)

= 1

Hence, option (2) is the answer.