Problem 32 - College Algebra

Question

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$ 2 x^{2}-6 x+7=0 $$

Step-by-Step Solution

Answer

The discriminant is -20, indicating two complex solutions.

Step 1: Identify the Coefficients

For the quadratic equation \(2x^2 - 6x + 7 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 2\), \(b = -6\), and \(c = 7\).

Step 2: Substitute Into Discriminant Formula

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Substitute the identified coefficients \(a = 2\), \(b = -6\), \(c = 7\) into this formula: \((-6)^2 - 4(2)(7)\).

Step 3: Calculate the Discriminant

Evaluate the discriminant: \(36 - 56 = -20\). Thus, the discriminant \(b^2 - 4ac = -20\).

Step 4: Determine the Number and Nature of Solutions

A discriminant less than 0 (here \(-20\)) indicates that the quadratic equation has two complex (non-real) solutions and no real solutions.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental component of algebra, typically expressed in the standard form \(ax^2 + bx + c = 0\). These equations can represent various natural phenomena, from projectile motion in physics to complex network flows in economics. Here, \(a\), \(b\), and \(c\) are coefficients which can shape the geometry and solutions of the equation.
To solve a quadratic equation, we might use various methods such as factoring, completing the square, or applying the quadratic formula. Each method serves to find the values of \(x\) that satisfy the equation, known as the roots.
One useful tool in this process is the **discriminant**, which helps determine the nature and number of solutions without solving the equation fully. The discriminant is calculated as \(b^2 - 4ac\), using the coefficients \(a\), \(b\), and \(c\). This particular equation, \(2x^2 - 6x + 7 = 0\), makes use of all these techniques in practical scenarios.

Exploring Complex Solutions

When discussing the roots of a quadratic equation, it’s crucial to understand the concept of complex solutions. A complex number includes a real part and an imaginary part, usually expressed as \(a + bi\), where \(i\) is the imaginary unit with the property \(i^2 = -1\).
Quadratic equations can yield complex solutions when the discriminant \(b^2 - 4ac\) is negative. This results in **non-real roots** because the square root of a negative number introduces the imaginary unit. In our example, \(b^2 - 4ac = -20\), clearly indicating complex roots.
Here’s how it breaks down: the calculation of \(\sqrt{-20}\) results in values involving \(i\), leading us to conclude the presence of two distinct complex solutions. These solutions are conjugates, ensuring they maintain a balanced structure in complex planes used to visualize them.

Nature of Solutions Indicated by the Discriminant

The nature of a quadratic equation's solutions is predominantly influenced by the discriminant \(b^2 - 4ac\). This critical value dictates whether the roots are real, repeated, or complex. Let's explore how:

• A **positive discriminant** means there are two distinct real solutions. This occurs when the parabola intersects the x-axis at two points.
• A **zero discriminant** indicates exactly one real solution, where the parabola touches the x-axis at the vertex.
• A **negative discriminant**, as found in our exercise with \(-20\), denotes that the solutions are complex or imaginary with no real intersection points on the graph.

Understanding these conditions helps predict the behavior of quadratic functions in graphical representations and in solving real-world mathematical problems.