How to Find the Roots of a Quadratic Equation

An equation of the form ax² + 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.

Here b² – 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.

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Solved Example

Example: If α and β are the roots of the equation x² – 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 the quadratic formula.

\(\begin{array}{l}x = \frac{ -1 \pm \sqrt{(1-4)}}{2}\end{array} \)

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.

Related Links:

• Quadratic Inequalities
• Previous Years Problems on Quadratic equations
• Linear equations

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