Quadratic equations are a fundamental concept in algebra and mathematics that are widely used in various fields, including physics, engineering, and economics. When we talk about a quadratic equation, we are referring to an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable we are solving for. In this article, we will delve into the roots of a quadratic equation and explore how to find the solutions of the quadratic equation 9x^2 – 6x – 2 = 0.

## Understanding Quadratic Equations

### What is a Quadratic Equation?

A **quadratic equation** is a second-degree polynomial equation in a single variable of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of the variable x in a quadratic equation is 2.

### Roots of a Quadratic Equation

The **roots** of a quadratic equation are the values of x that satisfy the equation, making it equal to zero. These roots are also known as the **solutions** or **zeros** of the quadratic equation. A quadratic equation can have two, one, or zero real roots based on the value of the discriminant (b^2 – 4ac).

### Finding the Roots of a Quadratic Equation

The roots of a quadratic equation can be determined by using the **quadratic formula**:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

## Solving 9x^2 – 6x – 2 = 0

Given the quadratic equation 9x^2 – 6x – 2 = 0, we can identify the values of a, b, and c as follows:

– a = 9

– b = -6

– c = -2

Substitute these values into the quadratic formula to find the roots of the equation:

x = (-(-6) ± sqrt((-6)^2 – 4 * 9 * (-2))) / 2 * 9

x = (6 ± sqrt(36 + 72)) / 18

x = (6 ± sqrt(108)) / 18

x = (6 ± 3√3) / 18

x = (2 ± √3) / 6

Therefore, the roots of the quadratic equation 9x^2 – 6x – 2 = 0 are:

x = (2 + √3) / 6 and x = (2 – √3) / 6

## FAQs about Quadratic Equations

### 1. What is the discriminant in a quadratic equation?

The **discriminant** in a quadratic equation is the expression b^2 – 4ac, which determines the nature of the roots. If the discriminant is positive, the equation has two distinct real roots; if it is zero, the equation has one real root; and if it is negative, the equation has two complex conjugate roots.

### 2. Can a quadratic equation have no real roots?

Yes, a quadratic equation can have **no real roots** if the discriminant is negative, leading to two complex conjugate roots that are not real numbers.

### 3. What is the role of the coefficient ‘a’ in a quadratic equation?

The **coefficient ‘a’** in a quadratic equation affects the shape and direction of the parabola represented by the equation. It determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0).

### 4. How are quadratic equations used in real-world applications?

Quadratic equations are commonly used to model various phenomena such as projectile motion, profit maximization in economics, and optimization problems in engineering and physics.

### 5. How can I check my solutions to a quadratic equation?

You can **verify** your solutions to a quadratic equation by substituting the values of x back into the original equation and confirming that it simplifies to zero. This ensures that the calculated roots are correct.

Understanding the roots of a quadratic equation is essential in mathematics and has practical applications in many fields. By mastering the principles behind quadratic equations, you can solve complex problems and analyze real-world scenarios with confidence.