Problem 47 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ x^{2}-\sqrt{5} x+1=0 $$

Step-by-Step Solution

Answer

The real solutions are \(x = \frac{\sqrt{5} + 1}{2}\) and \(x = \frac{\sqrt{5} - 1}{2}\).

Step 1: Identify the Quadratic Equation

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\). For this equation, \(a = 1\), \(b = -\sqrt{5}\), and \(c = 1\).

Step 2: Use the Quadratic Formula

To find the roots of the quadratic equation, we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the Discriminant

Calculate the discriminant, \(\Delta = b^2 - 4ac\). In this case, \(b = -\sqrt{5}\), so \(b^2 = 5\). Calculate \(\Delta = 5 - 4 \cdot 1 \cdot 1 = 1\).

Step 4: Find the Roots of the Equation

Substitute \(b = -\sqrt{5}\), \(\Delta = 1\), \(a = 1\) into the quadratic formula: \(x = \frac{-(-\sqrt{5}) \pm \sqrt{1}}{2 \times 1} = \frac{\sqrt{5} \pm 1}{2}\).

Step 5: Simplify the Result

The two real roots of the equation are \(x_1 = \frac{\sqrt{5} + 1}{2}\) and \(x_2 = \frac{\sqrt{5} - 1}{2}\).

Key Concepts

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of any quadratic equation, which is of the form \(ax^2 + bx + c = 0\). In this formula, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The quadratic formula itself is expressed as:

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

The "\(\pm\)" symbol indicates that there are generally two solutions, corresponding to the positive and negative roots. These are the two potential values of \(x\) that satisfy the original equation. By using this formula, we can directly calculate the solutions, also called roots, without factoring the quadratic equation.
Knowing the structure and application of the quadratic formula helps us approach problems systematically, ensuring that we don't miss any real root of a given quadratic equation.

Delving into the Discriminant

The discriminant is a key component of the quadratic formula and is represented by the expression \(b^2 - 4ac\). The discriminant effectively determines the nature and number of the roots of the quadratic equation.

• When the discriminant is positive (\(> 0\)), there are two distinct real solutions.
• If the discriminant is zero (\(= 0\)), there is exactly one real solution, also known as a repeated or double root.
• If the discriminant is negative (\(< 0\)), no real solutions exist; instead, the solutions are complex.

In our example, the discriminant is calculated as \(5 - 4 \cdot 1 \cdot 1 = 1\). Since this is positive, we confirm that there are two distinct real solutions for the quadratic equation given. Understanding the discriminant helps in predicting the number and type of the equation's roots without necessarily solving the entire equation.

The Nature of Real Solutions

Real solutions of a quadratic equation refer to the roots that are real numbers, which imply they can be plotted on the real number line. They are derived from the use of the quadratic formula, provided the discriminant is non-negative.
After calculating the roots using the quadratic formula with a positive discriminant, we obtain two solutions:

• \(x_1 = \frac{\sqrt{5} + 1}{2}\)
• \(x_2 = \frac{\sqrt{5} - 1}{2}\)

These are both real numbers, representing intersections of the graph of the equation \(x^2 - \sqrt{5}x + 1 = 0\) with the x-axis. Real solutions are valuable as they can be directly interpreted for practical applications like physics problems, geometry calculations, and more. Recognizing the nature of these solutions helps in understanding the behavior of quadratic graphs and functions in various contexts.