Problem 43 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The equation provided is a quadratic equation: \(x^2 - \sqrt{5}x + 1 = 0\). Our goal is to find the values of \(x\) that satisfy this equation.

Step 2: Identify the Coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = 1\), \(b = -\sqrt{5}\), and \(c = 1\). These coefficients will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \(x = \frac{-(-\sqrt{5}) \pm \sqrt{(-\sqrt{5})^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\): \((-\sqrt{5})^2 - 4 \cdot 1 \cdot 1 = 5 - 4 = 1\). The discriminant is 1, which means there are two distinct real solutions.

Step 5: Find the Roots

Using the quadratic formula, calculate the roots: \(x = \frac{\sqrt{5} \pm \sqrt{1}}{2}\). This simplifies to \(x = \frac{\sqrt{5} \pm 1}{2}\). Therefore, the two solutions are \(x = \frac{\sqrt{5} + 1}{2}\) and \(x = \frac{\sqrt{5} - 1}{2}\).

Key Concepts

Real Solutions

In mathematics, a real solution to an equation is a solution that is a real number. Real numbers include all the numbers on the number line without any imaginary components. When we solve a quadratic equation like \(x^2 - \sqrt{5}x + 1 = 0\), our focus is to find values of \(x\) that satisfy the equation within the real number system. When the quadratic equation has a discriminant greater than or equal to zero, it ensures that the solutions are real. For our exercise, the discriminant is 1, which clearly indicates that this equation has two real solutions. These solutions are\[ x = \frac{\sqrt{5} + 1}{2} \] and \[ x = \frac{\sqrt{5} - 1}{2} \]. They satisfy the equation without introducing any imaginary numbers.

Discriminant

The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. The discriminant is part of the quadratic formula, represented as \(b^2 - 4ac\). This expression helps us conclude whether the roots are real or complex:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution, sometimes called a repeated or double root.
• If the discriminant is negative, the solutions are complex or imaginary, meaning they do not appear on the real number line.

For the quadratic equation \(x^2 - \sqrt{5}x + 1 = 0\), our discriminant calculation \(5 - 4 = 1\) is positive, confirming two distinct real solutions for \(x\). This information is vital before solving any quadratic equation using the quadratic formula.

Quadratic Formula

The quadratic formula is a reliable method to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula can find the roots of any quadratic equation by simply substituting the values of the coefficients \(a\), \(b\), and \(c\). In our particular exercise, we had the coefficients \(a = 1\), \(b = -\sqrt{5}\), and \(c = 1\). By substituting these into the quadratic formula, the process allows you to:

• Compute the value of \(x\) using \(-b\), making it \(\sqrt{5}\) since \(-(-\sqrt{5}) = \sqrt{5}\).
• Then adding and subtracting the square root of the discriminant, here \(\sqrt{1}\), to get the two solutions.

Using this method ensures that you determine the roots systematically and accurately, as it incorporates every possibility of real, repeated, or complex solutions.

Coefficients

Coefficients in a quadratic equation are the numbers that multiply each term in the equation \(ax^2 + bx + c = 0\). Identifying these coefficients is the initial step in solving any quadratic equation. In our example, the equation \(x^2 - \sqrt{5}x + 1 = 0\) features the coefficients:

• \(a = 1\)
• \(b = -\sqrt{5}\)
• \(c = 1\)

These values are essential in calculations using the quadratic formula. The coefficient \(a\) represents the term associated with \(x^2\), \(b\) is paired with the linear term \(x\), and \(c\) is the constant. Understanding and identifying these correctly helps apply the quadratic formula appropriately, leading to the correct determination of the nature and values of the roots.