Problem 74 - A Graphical Approach to Precalculus with Limits

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 real, irrational solutions.

Step 1: Identify the coefficients

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

Step 2: Write the formula for the discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(\Delta = b^2 - 4ac\).

Step 3: Calculate the discriminant

Substitute the coefficients into the discriminant formula: \(\Delta = (-4)^2 - 4 \cdot 2 \cdot 1\). Calculate it: \(\Delta = 16 - 8 = 8\).

Step 4: Interpret the discriminant

Since \(\Delta = 8\) is greater than 0, the equation has two real solutions. Since \(\Delta\) is not a perfect square (it's 8), the solutions are irrational.

Key Concepts

Real Solutions

In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant helps us determine the number of real solutions the equation has. The discriminant, denoted as \( \Delta \), is calculated with the formula \( \Delta = b^2 - 4ac \). This value is crucial in assessing the nature of the solutions:

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

In the exercise provided, the discriminant was calculated to be 8. Since 8 is greater than 0, it confirms the presence of two real solutions for the quadratic equation \( 2x^2 - 4x + 1 = 0 \). Understanding this concept is essential for anyone studying quadratic equations as it simplifies the process of determining the nature of the roots without solving the equation explicitly.

Quadratic Formula

The quadratic formula is a powerful tool when it comes to solving quadratic equations. It provides the solutions for any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula itself is:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula is derived from completing the square on the quadratic equation. It allows students to find the roots of the equation directly by simply inputting the coefficients \( a \), \( b \), and \( c \).
A key part of the quadratic formula is the expression under the square root sign, which is the discriminant \( \Delta = b^2 - 4ac \). The value of the discriminant not only tells us if the solutions are real or complex but also whether they are rational or irrational. By assessing the discriminant, one can decide if using the quadratic formula is necessary or if the solutions can be determined by other means, such as factoring or simple calculation of square root if \( \Delta = 0 \). Understanding this formula is essential because it covers all possibilities and types of roots for quadratic equations.

Rational and Irrational Roots

Once the discriminant is known, it not only indicates the number of real solutions but also whether these solutions are rational or irrational.
The nature of the roots depends on whether \( \Delta \) is a perfect square:

• If \( \Delta \) is a perfect square (like 1, 4, 9, etc.), then the solutions are rational. This means they can be expressed as a fraction of two integers.
• If \( \Delta \) is not a perfect square, the solutions are irrational. These roots are still real, but they cannot be expressed as simple fractions and involve roots of non-perfect squares.

In our exercise, \( \Delta = 8 \), which is not a perfect square. This indicates that the quadratic equation \( 2x^2 - 4x + 1 = 0 \) has two irrational roots. Rational roots are often termed as more 'simple', while irrational roots may have a non-repeating, non-ending decimal expansion. Recognizing the type of roots can help anticipate the complexity of further calculations when working with quadratic equations.