Problem 71 - 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? \(x^{2}-3 x-7=0\)

Step-by-Step Solution

Answer

The discriminant of the equation \(x^{2}-3 x-7=0\) is 37, indicating that the equation has two distinct real solutions.

Step 1: Identify the coefficients

In the given equation \(x^{2}-3 x-7=0\), the coefficients can be identified as follows: \(a=1\), \(b=-3\), \(c=-7\).

Step 2: Calculate the Discriminant

Substitute the values of a, b, and c into the discriminant formula \(b^{2}-4ac\). Plug in known values: \((-3)^{2}-4*1*(-7)=9+28=37\). So, the discriminant, D, of the given equation is 37.

Step 3: Interpret the Discriminant's Value

Given that the discriminant D is 37, which is a positive number, this means that the quadratic equation will have two distinct real solutions. This is because for a quadratic equation, a positive discriminant implies two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a special type of polynomial equation that has a degree of 2. It can be written in the standard form:

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). Quadratic equations form a parabolic curve when graphed. This curve can either open upwards or downwards, depending on the sign of the coefficient \(a\). They can have either two, one, or no real solutions. The solutions to a quadratic equation are also called the roots of the equation. These roots can be found using different methods, such as factoring, completing the square, or using the quadratic formula.

Coefficients

Coefficients in a quadratic equation are the numbers that multiply the variables. In a quadratic equation like \(ax^2 + bx + c = 0\):

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term, not attached to a variable

In our example equation, \(x^2 - 3x - 7 = 0\):

• The coefficient \(a = 1\)
• The coefficient \(b = -3\)
• The constant term \(c = -7\)

Understanding these coefficients is crucial as they play vital roles in determining the graph's shape and the equation's solutions.

Real Solutions

The solutions of a quadratic equation can either be real or complex numbers. Real solutions occur when the values at which the quadratic equation equals zero are real numbers. These solutions are the points where the graph of the quadratic equation intersects the x-axis.

For a quadratic equation to have real solutions, its discriminant must be greater than or equal to zero. There are two scenarios:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, which means the graph touches the x-axis at one point. This is also known as a repeated or double root.

Understanding whether a quadratic equation has real solutions is a key part of analyzing its behavior and how its graph looks.

Discriminant Interpretation

The discriminant of a quadratic equation provides valuable insight into the nature of its roots. It is calculated using the formula \(D = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation. The discriminant helps us determine:

• If \(D > 0\), the equation has two distinct real solutions. This indicates that the parabola intersects the x-axis at two points.
• If \(D = 0\), there is exactly one real solution, indicating that the parabola touches the x-axis at exactly one point without crossing it.
• If \(D < 0\), there are no real solutions, implying that the parabola does not intersect the x-axis. Instead, it has two complex conjugate solutions.

For the given equation, \(x^2 - 3x - 7 = 0\), the discriminant value is 37, indicating two distinct real solutions. This positive discriminant means we are dealing with a graph that cuts through the x-axis at two separate points.