Problem 75 - College Algebra Essentials

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 is 36. Therefore, the quadratic equation \(x^{2}-4x-5=0\) has two distinct real roots.

Step 1: Identify the coefficients

In the equation \(x^{2}-4x-5=0\), the coefficient of \(x^{2}\) (a) is 1, the coefficient of x (b) is -4, and the constant term (c) is -5.

Step 2: Compute the discriminant

Use the discriminant formula \(D = b^{2} - 4ac\). Substituting the identified values, we get\(D = (-4)^{2} - 4*1*(-5) = 16 + 20 = 36\).

Step 3: Determine the nature of the roots

Since the discriminant D is greater than 0, the equation has two distinct real roots. This is concluded from the fact that if \(D > 0\), the quadratic equation will have two distinct real roots.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are polynomial equations of degree two, which generally appear in the standard form: \( ax^2 + bx + c = 0 \). Here:

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

The value of \(a\) must not be zero, as that would make the equation linear instead of quadratic.
Quadratic equations are fundamental in algebra and arise in numerous mathematical and real-world contexts. They can represent different geometric shapes, such as parabolas, when graphed in a coordinate system. The solutions to these equations, also known as roots, are found using different techniques, one of the key ones being the quadratic formula.The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), offers a straightforward way to find the roots of any quadratic equation once you have the coefficients \(a\), \(b\), and \(c\).
Crucially, the part under the square root sign, termed the discriminant \((b^2 - 4ac)\), plays a vital role in helping us understand the nature of these solutions.

What Are Real Roots?

Real roots of a quadratic equation are the values of \(x\) that satisfy the equation when it is set to zero.
These real roots are simply the solutions that you can plot on a number line. For a quadratic equation, the roots can appear in two forms:

• Distinguished real roots: These occur when the discriminant \(D\) is positive \((D > 0)\). It implies that the parabola intersecting the x-axis at two distinct points, offering two different real solutions.
• Double or repeated root: If the discriminant \(D\) equals zero \((D = 0)\), this means the parabola just touches the x-axis at one point, indicating a single real root that repeats.

The concept of real roots is significant because it not only describes the intercepts of the graph of the quadratic equation but also impacts the overall shape and behavior of the parabola. In practical problem-solving scenarios, knowing whether a quadratic equation has real or non-real roots can significantly affect the approach and solution strategy.

The Nature of Solutions of Quadratic Equations

The nature of solutions in quadratic equations is primarily determined by the discriminant \(D = b^2 - 4ac\). This value holds the key to understanding how many and what type of solutions you can expect.
Here's a quick guide:- **Positive Discriminant \((D > 0)\):** The equation has two distinct real roots. An example is when the graph of the equation intersects the x-axis at two separate points.- **Zero Discriminant \((D = 0)\):** There is exactly one real root because the parabola barely touches the x-axis at a single point. This is also called a repeated or double root.- **Negative Discriminant \((D < 0)\):** The equation has no real roots but instead has two complex conjugate roots, which means the graph does not cross the x-axis at all.Understanding the discriminant allows students to quickly determine the number and type of roots without actually solving the equation fully.
In the given example \(x^2 - 4x - 5 = 0\), the discriminant is 36, a positive number, therefore it has two distinct real roots. This insight aids in drawing and predicting the behavior of the solution graph.