Problem 36 - 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. $$ 2 x^{2}-3 x-7=0 $$

Step-by-Step Solution

Answer

The discriminant is 65; there are two distinct real solutions.

Step 1: Identify the Coefficients

First, identify the coefficients in the quadratic equation. The given quadratic equation is \(2x^2 - 3x - 7 = 0\). Here, \(a = 2\), \(b = -3\), and \(c = -7\).

Step 2: Write the Discriminant Formula

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

Step 3: Substitute the Coefficients into the Discriminant Formula

Substitute \(a = 2\), \(b = -3\), and \(c = -7\) into the discriminant formula: \(D = (-3)^2 - 4(2)(-7)\).

Step 4: Calculate the Discriminant

Calculate the discriminant: \(-3^2 = 9\) and \(-4 \times 2 \times -7 = 56\). So, \(D = 9 + 56 = 65\).

Step 5: Determine the Nature and Number of Solutions

Since the discriminant \(D = 65\) is greater than zero, the quadratic equation has two distinct real solutions.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a type of polynomial equation where the highest power of the variable, typically denoted as \(x\), is 2. It generally appears in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients:

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\), and
• \(c\) is the constant term.

This structure means it creates a parabolic graph when plotted, forming a curve. It can open upwards if \(a > 0\), or downwards if \(a < 0\). Quadratic equations are used in various fields, including physics and engineering, to model various situations where relationships are not linear. They help determine values like projectile paths or areas. Recognizing the basic format is essential before analyzing the nature of its solutions.

Exploring Real Solutions in Quadratic Equations

Real solutions are the solutions of a quadratic equation that are real numbers. To determine the nature of real solutions for a quadratic equation, you use the discriminant, \(D\), calculated as \(b^2 - 4ac\). The discriminant offers insights into how many real solutions exist and what they might look like:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution or root, often called a repeated or double root.
• If \(D < 0\), the solutions are not real numbers; instead, they are complex (involving imaginary numbers).

For our specific equation \(2x^2 - 3x - 7 = 0\), we calculated \(D = 65\). Since 65 is greater than zero, this equation has two distinct real solutions. This understanding lets us know that the parabola intersects the x-axis at two points.

Evaluating the Nature of Solutions Using the Discriminant

The discriminant of a quadratic equation not only tells us the number of solutions but also reveals the nature of these solutions. The nature of solutions refers to whether solutions are real, repeated, or whether they are complex.

• Two distinct real solutions occur because the parabola associated with the quadratic equation intersects the x-axis at two points. This is the case when \(D > 0\). Between these intersections is where your variable takes real values.
• A single solution (\(D = 0\)) implies the parabola touches the x-axis at exactly one point, known as the vertex of the parabola. This indicates a double root situation.
• When there are no real solutions (\(D < 0\)), the parabola doesn't intersect the x-axis at all, indicating that the solutions are complex and occur as a pair of complex conjugates.

The discriminant is a valuable part of understanding a quadratic equation because it outlines the potential outcomes and the underlying properties of the solutions without actually solving the equation itself. This makes it a powerful tool in algebra and critical for solving equations more efficiently.