Problem 72 - A Graphical Approach to College Algebra

Question

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$2 x^{2}-4 x+1=0$$

Step-by-Step Solution

Answer

Two irrational real solutions.

Step 1: Identify Coefficients

The given quadratic equation is \(2x^2 - 4x + 1 = 0\). Identify the coefficients: \(a = 2\), \(b = -4\), \(c = 1\).

Step 2: Calculate the Discriminant

Use the formula for the discriminant: \(\Delta = b^2 - 4ac\). Substitute the identified coefficients: \(\Delta = (-4)^2 - 4 \times 2 \times 1 = 16 - 8 = 8\).

Step 3: Analyze the Discriminant

Since the discriminant \(\Delta = 8\) is positive, the quadratic equation has two real solutions. Because 8 is not a perfect square, the solutions are irrational.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree, which means that the highest power of the variable, usually represented as \(x\), is 2. It typically takes the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This form is called the *standard form* of a quadratic equation.
Understanding the coefficients in a quadratic equation is crucial, as they play significant roles in determining the nature of the solutions.

• The coefficient \(a\) is associated with the term \(x^2\) and cannot be zero, or else it wouldn't be a quadratic equation.
• The coefficient \(b\) is the linear term coefficient.
• The constant \(c\) is the free term.

The solutions to quadratic equations are typically found using methods like factoring, completing the square, or using the quadratic formula. The discriminant, which we will discuss next, helps determine the nature of these solutions.

Real Solutions

In the context of quadratic equations, real solutions are the values of \(x\) that satisfy the equation and are real numbers. The discriminant, represented as \(\Delta = b^2 - 4ac\), is used to determine whether a quadratic equation has real solutions. Here's how it works:

• If \(\Delta > 0\), the equation has two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution, also called a repeated or double root.
• If \(\Delta < 0\), the equation has no real solutions but rather two complex solutions.

In our specific example, the discriminant is \(\Delta = 8\), which is greater than zero. This indicates the quadratic equation \(2x^2 - 4x + 1 = 0\) has two distinct real solutions. Note that the specific value of the discriminant, in this case, does not only help identify the number of solutions but also hints at their specific nature, as discussed in the next section.

Rational and Irrational Solutions

Real solutions can be further categorized into rational and irrational solutions, depending on the nature of the discriminant's square root.

• Rational solutions occur when the square root of the discriminant is a rational number, i.e., it can be expressed as a fraction of two integers.
• Irrational solutions occur when the square root of the discriminant is not a perfect square, implying that it leads to non-repeating, non-terminating decimals.

In the given example, the discriminant is \(\Delta = 8\). Since 8 is not a perfect square, the square root of 8 is an irrational number. Hence, the solutions to the equation \(2x^2 - 4x + 1 = 0\) are irrational. Identifying whether the solutions are rational or irrational helps in understanding the precision and nature of the solution set.