Problem 75 - College Algebra

Question

In Exercises \(75-82\), compute the discriminant. Then determine the number and type of solutions for the given equation. $$x^{2}-4 x-5=0$$

Step-by-Step Solution

Answer

The discriminant of the given quadratic equation is 36 which indicates that the equation has two distinct real solutions.

Step 1: Find the Discriminant

Use the formula for computing the discriminant, which is \(D = b^{2} - 4ac\). Here, \(a = 1, b = -4 \)and \(c = -5\). Plug these values into the formula to find the discriminant.

Step 2: Calculate the Discriminant

After substitution, we get \[D = (-4)^{2} - 4(1)(-5) = 16 + 20 = 36\].

Step 3: Determine the Nature of Roots

Since the discriminant is more than zero (in this case, 36), the quadratic equation, \( x^{2} - 4x - 5 = 0\), has two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. The general form of a quadratic equation is:

• \( ax^{2} + bx + c = 0 \)

where \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). The variable \(x\) represents an unknown value we aim to solve for.
Finding the solution requires identifying the roots of the equation, meaning the values of \(x\) that satisfy the equation. To solve a quadratic equation, you can use several methods such as factoring, completing the square, or applying the quadratic formula.
Among these, the quadratic formula is particularly useful because it can solve any quadratic equation:

• \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

The term under the square root, \(b^{2} - 4ac\), is known as the discriminant.

Real Solutions

When solving a quadratic equation, understanding the discriminant is key to determining the number of real solutions. The discriminant is expressed as:

• \(D = b^{2} - 4ac\)

Depending on the value of \(D\), a quadratic equation can have either real or complex solutions.
Here’s how you can interpret the discriminant:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), the equation has exactly one real solution, also known as a repeated or double root.
• If \(D < 0\), the equation has no real solutions, but two distinct complex solutions.

In the given equation \(x^{2} - 4x - 5 = 0\), the discriminant \(D\) is 36, indicating there are two distinct real solutions.

Nature of Roots

The nature of roots provides insight into the behavior or characteristics of the solutions of a quadratic equation. By examining the discriminant, you can anticipate what kind of solutions to expect without solving the equation completely.
Below is a breakdown based on the value of the discriminant:

• Positive Discriminant (\(D > 0\)): Two distinct real roots are present. The graph of the quadratic function will intersect the x-axis at two points.
• Zero Discriminant (\(D = 0\)): One real double root exists. This means the parabola touches the x-axis at one point, indicating a vertex on the x-axis.
• Negative Discriminant (\(D < 0\)): No real roots, but two distinct complex roots exist. The graph of the quadratic function does not intersect the x-axis at any point.

In the exercise, we determined that \(D = 36\), a positive number, leads to two distinct real roots, reflecting that the equation's graph intersects the x-axis at two separate points.