Problem 68 - College Algebra

Question

Compute the discriminant of each equation in Exercises 65-72 What does the discriminant indicate about the number and type of solutions? \(2 x^{2}+11 x-6=0\)

Step-by-Step Solution

Answer

The discriminant of the equation \(2x^2 + 11x - 6 = 0\) is 169. Since the discriminant is positive, the equation has two distinct real roots.

Step 1: Identifying the coefficients

Our given equation is \(2x^2 + 11x - 6 = 0\). In this case, \(a = 2\), \(b = 11\), and \(c = -6\).

Step 2: Substitution into the Discriminant formula

We substitute a, b, and c into the discriminant formula \(D = b^2 - 4ac\). In this case, \(D = (11)^2 - 4 * 2 * -6\).

Step 3: Calculate the Discriminant

Now, calculate the discriminant value: \(D = 121 - (-48)\), which yields \(D = 169\).

Step 4: Analyze the Discriminant

As the discriminant is greater than zero, there will be two distinct real roots for the quadratic equation.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. This specific quadratic structure creates a parabola when graphed on a coordinate plane. The solutions to these equations, also known as roots, are the points at which the graph intersects the x-axis.

For example, to solve the quadratic equation presented in the exercise, \(2x^2 + 11x - 6 = 0\), we look for values of \(x\) that make this equation true. These solutions could be real numbers or complex numbers, and their nature can be determined using discriminant analysis, which we'll discuss in the next section.

Discriminant Analysis

The discriminant is a powerful tool in algebra that helps us analyze the nature of the roots of a quadratic equation without actually solving the equation. It is represented by \(D\) and is calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).

In the given example, the discriminant was calculated as follows: \(D = (11)^2 - 4 \times 2 \times -6 = 121 - (-48) = 169\). The value of the discriminant, 169, is positive, which tells us there are two distinct real roots. Had the discriminant been zero, there would be exactly one real root (a repeated root), and if the discriminant were negative, the quadratic equation would have two complex roots. These different outcomes based on the discriminant give us a quick and clear insight into the solutions of the quadratic equation.

Real Roots of Polynomial

When dealing with quadratic equations, the term 'real roots' refers to the solutions that are real numbers. A quadratic equation can have either zero, one, or two real roots depending on the discriminant's value. If the discriminant is positive, as in the example \(2x^2 + 11x - 6 = 0\), with \(D = 169\), there are two separate real roots. These roots are the x-intercepts of the parabola represented by the quadratic equation.

The fact that the discriminant in our example is a perfect square further indicates that the roots are rational numbers. When you solve for the roots using the quadratic formula \( x = \frac{-b \bpm \b sqrt{D}}{2a} \), you will indeed find two rational solutions. These are the points where the parabola cuts the x-axis, graphically representing where the equation equals zero. Understanding the connection between the discriminant and the nature of the roots greatly simplifies comprehending the behavior of polynomials.