91Ó°ÊÓ

When tackling a quadratic equation like the one provided: \(-3.22 x^{2}-0.08 x+28.651=0\), the first step is almost always identifying the correct method for solution. In this case, we're using the quadratic formula, which is an elegant, universally applicable approach whenever the equation is restructured into its standard form of \(ax^2 + bx + c = 0\).

As you've learned, the quadratic formula is \(\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), which acts as a failsafe mechanism, capable of finding the roots of any quadratic equation, given that you can identify and substitute the values of \(a\), \(b\), and \(c\)—the coefficients of the equation—correctly.

In using this method, it's important to follow the steps concisely: identify coefficients, substitute into the quadratic formula, perform the operations under the square root, simplify, and solve for the roots (or solutions). Carrying out these steps with care and precision leads to the two possible solutions for the variable \(x\).

The coefficients in a quadratic equation are the numerical parts that stand in front of the \(x^2\), \(x\), and the constant term. In the equation \(-3.22x^{2} - 0.08x + 28.651 = 0\), the coefficients are crucial for solving the equation using the quadratic formula.

These coefficients are labeled as \(a = -3.22\) for the \(x^2\) term, \(b = -0.08\) for the \(x\) term, and \(c = 28.651\) for the constant term. Each plays a distinct role in determining the curvature and position of the parabola that the quadratic equation represents when graphed.

Correct Identification of Coefficients

Ensuring correct identification of these coefficients is essential for the effectiveness of the quadratic formula. Remember, a small error in this initial step can lead to incorrect solutions. The real power of understanding coefficients becomes evident when you realize how they influence the nature and position of roots on the graph.

The roots (or solutions) of a quadratic equation are the values of \(x\) at which the equation equals zero. Geometrically, they represent the points where the parabola, which is the graph of the quadratic function, intersects the x-axis.

From the exercise, after using the quadratic formula, we arrive at two roots: \(x_1 = 2.712\) and \(x_2 = -3.323\), rounded to three decimal places. These roots can be real and distinct, real and equal, or even complex (containing imaginary numbers), depending on the discriminant—\(b^2 - 4ac\).

Understanding the Discriminant

The discriminant in the quadratic formula gives insight into the nature of the roots without even computing them: if positive, there are two distinct real roots; if zero, there is one real root (a double root); and if negative, the roots are complex. In this exercise, the discriminant was a positive number, indicating that the quadratic had two distinct real roots. Recognizing the importance of the discriminant helps in anticipating the number and type of solutions, which is a valuable skill for algebraic problem solving.