One root of the equation \(3 x^{2}-2 x-2=0\) is a) \(1.215\) b) \(1.064\) c) \(0.937\) d) \(0.826\)

Understanding the quadratic formula is like having a master key for solving quadratic equations. At its heart, the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is derived from the process of completing the square, and it provides a reliable method to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \).

The beauty of this formula lies in its universality—no matter how complex the coefficients, as long as they are real numbers, the quadratic formula will find the roots. The symbols \( \pm \) indicate that there are usually two solutions, known as roots, to a quadratic equation. These roots can sometimes be real and distinct, real and equal, or complex numbers, depending on the discriminant \( b^2 - 4ac \).

The discriminant is the part of the quadratic formula under the square root: \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of a quadratic equation. A positive discriminant means two real and distinct roots, zero means one real repeated root, and a negative discriminant indicates that the roots are complex and come in conjugate pairs.

For the given equation \( 3 x^{2}-2 x-2=0 \), the discriminant is calculated as \( D = (-2)^2 - 4(3)(-2) = 28 \), which is positive, confirming the existence of two real and distinct roots. It's the discriminant that gives us critical information about the roots even before calculating them.

Solving quadratic equations is essentially the process of finding values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). There are several methods for solving these equations, such as factoring, completing the square, or graphing. However, the most widely used method is the application of the quadratic formula because it is applicable in all cases and is relatively straightforward.

After determining the coefficients \( a \), \( b \), and \( c \) and the discriminant, simply plug them into the quadratic formula to find the roots. In our exercise example, by calculating and substituting \( a = 3 \)\, \( b = -2 \)\, and \( c = -2 \), we receive two roots which we can then simplify to find the approximate values.

The coefficients in a quadratic equation are the numerical multipliers of the terms \( ax^2 \)\, \( bx \)\, and \( c \) in the standard quadratic form \( ax^2 + bx + c = 0 \). These coefficients directly influence the shape and position of the equation's graph on a coordinate plane. In our exercise, the coefficient \( a = 3 \) impacts the concavity and width of the parabola, \( b = -2 \) affects the horizontal placement, and the constant term \( c = -2 \) shifts the graph vertically.

These coefficients are the starting point for solving the quadratic equation, impacting both the discriminant calculation and the resultant roots obtained through the quadratic formula. Identifying them correctly is a crucial step in the process.