Problem 36 - College Algebra

Question

Determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve. $$2 x^{2}-3 x-7=0$$

Step-by-Step Solution

Answer

Two distinct real solutions (discriminant = 65).

Step 1: Identify coefficients

First, identify the coefficients in the quadratic equation in the form \( ax^2 + bx + c = 0 \). For the equation \( 2x^2 - 3x - 7 = 0 \), we have:\\( a = 2 \)\\( b = -3 \)\\( c = -7 \).

Step 2: Use the discriminant formula

The discriminant \( \, \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( b^2 - 4ac \). Substitute the values we identified: \\( \Delta = (-3)^2 - 4(2)(-7) = 9 + 56 = 65 \).

Step 3: Analyze the discriminant value

A positive discriminant indicates two distinct real solutions, a discriminant of zero indicates exactly one real solution, and a negative discriminant indicates two complex solutions. Since \( \Delta = 65 \) is positive, the equation has two distinct real solutions.

Key Concepts

Understanding the Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two, meaning the highest exponent of the variable (usually represented as \( x \)) is squared, or raised to the power of two. These equations can be written in the standard form: \( ax^2 + bx + c = 0 \). Each letter \( a \), \( b \), and \( c \) in this equation represents coefficients that determine the specific shape and position of the parabola when graphed.
To solve a quadratic equation, one common method is using the discriminant. The discriminant offers insight into the type and number of solutions—"roots"—that the equation has without actually solving it. This is achieved through the discriminant formula, \( \Delta = b^2 - 4ac \). By identifying the values of \( a \), \( b \), and \( c \), students can plug these into the discriminant formula to determine the nature of the solutions for any quadratic equation.

Exploring Real Solutions

Real solutions to a quadratic equation are those solutions that are real numbers. These solutions can be found when the discriminant is zero or positive. These real solutions relate to actual points where the parabola intersects or touches the x-axis on a graph.

There are two scenarios:

• If the discriminant \( \Delta \) is positive, like in the original exercise where \( \Delta = 65 \), the equation has two distinct real solutions. This indicates that the parabola intersects the x-axis at two distinct points.
• If the discriminant is zero, the equation has exactly one real solution, often called a repeated root. It means the parabola tangentially "touches" the x-axis at just one point, creating only one intersection point.

Understanding these scenarios helps in predicting the behavior of quadratic equations and their graphs without solving them completely.

Delving into Complex Solutions

Complex solutions come into play when the discriminant of a quadratic equation is negative. This scenario implies that the parabola does not intersect the x-axis at any real point because the roots are not real numbers.

Complex numbers include imaginary numbers and are typically expressed in the form \( a + bi \) where \( i \) is the imaginary unit square root of \(-1\). For a quadratic equation with a negative discriminant, the solutions will always occur in conjugate pairs, such as \( a + bi \) and \( a - bi \).

When graphing such an equation, the parabola will be entirely above or below the x-axis, never touching or crossing it. This characteristic signifies that all solutions involve imaginary numbers, providing an essential insight for problems that extend beyond real number calculations and delve into complex analysis.