Problem 37 - College Algebra with Corequisite Support

Question

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 6 x^{2}-x-2=0 $$

Step-by-Step Solution

Answer

Two distinct real solutions.

Step 1: Identify the Coefficients

The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). For the equation \( 6x^2 - x - 2 = 0 \), identify the coefficients: \( a = 6 \), \( b = -1 \), and \( c = -2 \).

Step 2: Write the Discriminant Formula

The formula for the discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). This formula will help determine the nature of the solutions.

Step 3: Calculate the Discriminant

Using the values \( a = 6 \), \( b = -1 \), and \( c = -2 \), substitute them into the discriminant formula: \[ D = (-1)^2 - 4(6)(-2) \]. Calculate each part: \((-1)^2 = 1\) and \(4 \times 6 \times 2 = 48\). The discriminant \( D = 1 + 48 = 49 \).

Step 4: Interpret the Discriminant

Since the discriminant \( D = 49 \) is a positive perfect square, the quadratic equation has two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \). It is called "quadratic" because the highest degree of its variable \( x \) is two.

In the given exercise, the quadratic equation is \( 6x^2 - x - 2 = 0 \). Identifying the coefficients is crucial: \( a = 6 \), \( b = -1 \), and \( c = -2 \). These represent the numerical factors for the terms in the equation and guide further analysis, such as calculating the discriminant.

Understanding quadratic equations involves recognizing their standard form and how different components contribute to determining the nature of their solutions.

Real Solutions

The solutions to a quadratic equation such as \( ax^2 + bx + c = 0 \) can be found using several methods, like factoring or applying the quadratic formula. However, knowing the discriminant helps predict the nature of these solutions without solving them completely.

For quadratic equations, solution types can be classified as follows:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, sometimes called a double root.
• If the discriminant is negative, there are no real solutions, but two complex ones.

In our example, the discriminant is a positive perfect square \( D = 49 \), so the equation \( 6x^2 - x - 2 = 0 \) possesses two distinct real solutions. This means that the graph of the quadratic function intersects the x-axis at two different points.

Discriminant Formula

The discriminant is a key concept in understanding the solutions to a quadratic equation, derived from the quadratic formula \( x = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a} \). Only the part under the square root, \( b^2 - 4ac \), known as the discriminant, is needed to determine the number and types of solutions.

Let's explore the steps:

• First, recall the coefficients: here, \( a = 6 \), \( b = -1 \), and \( c = -2 \).
• Use the formula: \( D = b^2 - 4ac \).
• Substitute: \( D = (-1)^2 - 4(6)(-2) = 1 + 48 = 49 \).

In conclusion, the positive value of \( D = 49 \) indicates two real solutions. This thorough understanding of the discriminant is crucial when number-crunching through quadratic equations.