Problem 1 - College Algebra and Calculus: An Applied Approach

Question

In Exercises 1-8, use the discriminant to determine the number of real solutions of the quadratic equation. \(4 x^{2}-4 x+1=0\)

Step-by-Step Solution

Answer

The quadratic equation \(4x^2 - 4x + 1 = 0\) has exactly one real solution.

Step 1: Identify the coefficients

First, identify the coefficients of the quadratic equation \(4x^2 - 4x + 1 = 0\). Here, \(a = 4\), \(b = -4\), and \(c = 1\).

Step 2: Calculate the Discriminant

Next, calculate the discriminant \(D\), which is given by the formula \(D = b^2 - 4ac\). Substituting the identified coefficients: \(D = (-4)^2 - 4*4*1 = 16 - 16 = 0\).

Step 3: Determine the number of real solutions

Finally, determine the number of real solutions. Since the discriminant \(D = 0\), the quadratic equation has exactly one real solution.

Key Concepts

Real Solutions of Quadratic Equations

Understanding the nature of solutions to quadratic equations is crucial for various mathematical applications. Quadratic equations are often in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown. The solutions to these equations can be real or complex numbers.

When we talk about real solutions, we mean the values of \(x\) that are not accompanied by the imaginary unit \(i\). The number of real solutions depends directly on the value of the discriminant, which we'll discuss in the following sections. A positive discriminant indicates two distinct real solutions, a zero discriminant points to exactly one real solution (also known as a repeated or double root), and a negative discriminant tells us that there are no real solutions, just complex ones.

Calculating Discriminant

The discriminant is a powerful tool used to determine the nature and number of solutions of a quadratic equation without actually solving the equation. The discriminant, often denoted as \(D\), is part of the quadratic formula and can be computed using the coefficients \(a\), \(b\), and \(c\) of the quadratic equation.

It is calculated with the formula \(D = b^2 - 4ac\). The outcome of this calculation will dictate whether the quadratic equation has two real solutions (if \(D > 0\)), one real solution (if \(D = 0\)), or two complex solutions (if \(D < 0\)). For example, with an equation \(4x^2 - 4x + 1 = 0\), using the coefficients, the discriminant would be \(D = (-4)^2 - 4(4)(1)\), which simplifies to \(D = 0\). This tells us there is exactly one real solution.

Discriminant Case Study

• If \(D > 0\), there are two real and distinct solutions.
• If \(D = 0\), there is one real and repeated solution.
• If \(D < 0\), there are no real solutions, only two complex ones.

Quadratic Formula

The quadratic formula is perhaps the most recognized tool for solving quadratic equations and it directly incorporates the discriminant. This formula provides the solutions for \(x\) in any quadratic equation \(ax^2 + bx + c = 0\): \(x = \frac{-b \pm \sqrt{D}}{2a}\), where \(\pm\) signifies that there are usually two solutions, depending on the sign chosen.

Applying this formula involves inserting the identified coefficients and the computed discriminant. Going back to our example, \(4x^2 - 4x + 1 = 0\), we found that \(D = 0\). Using the quadratic formula with our coefficients and this discriminant gives us the real solution: \(x = \frac{-(-4) \pm \sqrt{0}}{2(4)} = 1\), confirming that there is one real solution at \(x = 1\).

Application of the Quadratic Formula

• It systematically solves any quadratic equation.
• Regardless of the discriminant's value, the formula applies: positive, zero, or negative.
• It reveals the actual solutions after determining the nature of the solutions via the discriminant.