The Mysteries Behind 3x2 - 1 0: Beyond Just a 3, b 0, and c -1

When dealing with the quadratic equation 3x2 - 1 0, it's pivotal to understand the components that make up this equation. The coefficients a, b, and c play crucial roles in the analysis and solving of such equations. By rewriting the equation in the standard form ax2 bx c 0, we can easily identify these coefficients.

Identifying Coefficients a, b, and c

For the given equation 3x2 - 1 0:

When written in the standard form, we have: a 3, the coefficient of x2 b 0, as there is no x term c -1, the constant term

Understanding the Standard Form Polynomial

A standard form polynomial is given as ax2 bx c 0. If we take another example, the equation x2 - 3x - 2 0 would have:

a 1 b -3 c -2

This shows how the coefficients are derived from the polynomial. Without any x term, b is assumed to be 0.

Further Exploration of Quadratic Equations

Beyond just identifying the coefficients, there are several interesting aspects of quadratic equations worth discussing:

Roots of a Quadratic

The roots of a quadratic equation can be found using the quadratic formula:

x u00BD(-b ± u221A(b2 - 4ac))

For 3x2 - 1 0, the roots can be calculated as:

x u00BD(0 ± u221A(02 - 4 * 3 * -1)) u00BD(0 ± u221A12) ± u221A(12)/2 ± u221A(3)

Axis of Symmetry

The axis of symmetry for a quadratic equation is given by -b/2a. For 3x2 - 1 0:

Axis of symmetry -0/(2 * 3) 0

This means the curve is symmetric about the y-axis.

Sum and Product of Roots

The sum of the roots is -b/a, and the product of the roots is c/a. For 3x2 - 1 0: Sum of roots -0/3 0 Product of roots -1/3

Graphing the Quadratic Equation

The graph of a quadratic equation forms a parabola. The key features include:

The vertex can be found using the axis of symmetry. For 3x2 - 1 0, the vertex is at (0, -1). The directrix and foci can be determined using the properties of a parabola. However, these are advanced concepts not typically covered in basic algebra. The parabola is concave up because the leading coefficient (a) is positive.

Differentiation to Find the Gradient

To find the gradient of the parabola at any point, we can take the derivative of the equation:

d/dx (3x2 - 1) 6x

This derivative, 6x, represents the slope of the tangent line at any point on the parabola.

Area Bounded by the Curve

The area bounded by the curve can be found using the definite integral:

Area u222B (-1/u221A3 to 1/u221A3) (3x2 - 1) dx

By evaluating this integral, we can determine the area under the curve between the roots.

Transformations from the Parent Function

The standard quadratic equation y ax2 bx c can be transformed:

A factor by 3 changes the equation to: y 3x2 - 3x - 3 Move 1 unit down changes the equation to: y 3x2 - 3x - 4

These transformations are useful for graphing and understanding the behavior of the quadratic function.

Thus, while the coefficients a, b, and c are fundamental, there is a lot more to a quadratic equation than just finding these values. Understanding the roots, axis of symmetry, and other properties can provide deeper insights into the behavior and characteristics of quadratic functions.