Problem 35 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ x^{2}+3 x+1=0 $$

Step-by-Step Solution

Answer

The real solutions are \( x = \frac{-3 + \sqrt{5}}{2} \) and \( x = \frac{-3 - \sqrt{5}}{2} \).

Step 1: Identify the Type of Equation

The given equation is a quadratic equation: \( x^2 + 3x + 1 = 0 \). We need to find its roots using the quadratic formula, since it may not factorize easily.

Step 2: Recall the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), used for solving equations of the form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = 3 \), and \( c = 1 \).

Step 3: Calculate the Discriminant

Calculate the discriminant using \( b^2 - 4ac \). Substituting the known values, we have: \( 3^2 - 4 \times 1 \times 1 = 9 - 4 = 5 \). Since the discriminant is positive, there are two distinct real roots.

Step 4: Apply the Quadratic Formula

Substitute \( a \), \( b \), and \( c \) into the quadratic formula: \( x = \frac{-3 \pm \sqrt{5}}{2} \).

Step 5: Simplify the Expression

Compute the two possible solutions: \( x = \frac{-3 + \sqrt{5}}{2} \) and \( x = \frac{-3 - \sqrt{5}}{2} \). These are the real solutions to the equation.

Key Concepts

Understanding the Quadratic Formula

Quadratic equations like the one given, \(x^2 + 3x + 1 = 0\), appear often in algebra. The quadratic formula is a robust tool for finding the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). When you have coefficients \(a\), \(b\), and \(c\), you can find the solutions for \(x\) using:

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

This formula provides a straightforward path to the roots, whether or not the equation can be factored easily. Here, the coefficients are \(a = 1\), \(b = 3\), and \(c = 1\). The formula accounts for every possibility, from equations with two distinct roots to cases where a solution might repeat.

The Role of the Discriminant

The discriminant is the part of the quadratic formula under the square root: \(b^2 - 4ac\). It plays a crucial role in determining the nature of the roots of a quadratic equation. Here's how:

• If the discriminant is positive, as in this case \(3^2 - 4 \times 1 \times 1 = 5\), it indicates that the quadratic equation has two distinct real solutions.
• If the discriminant equals zero, there is exactly one real solution, and the root is repeated.
• If the discriminant is negative, the quadratic equation has no real solutions; instead, you'll get two complex solutions.

Thus, in the process of solving \(x^2 + 3x + 1 = 0\), the positive discriminant \(5\) means we will find two distinct real solutions.

Finding Real Solutions

Real solutions to a quadratic equation can be explicitly calculated using the quadratic formula once the discriminant has been evaluated. As seen, the given equation is solved as follows:
Plug in \(a = 1\), \(b = 3\), and the discriminant \(\sqrt{5}\) into the quadratic formula:

• \(x = \frac{-3 \pm \sqrt{5}}{2}\)

This yields the two separate solutions:

• \(x = \frac{-3 + \sqrt{5}}{2}\)
• \(x = \frac{-3 - \sqrt{5}}{2}\)

These solutions are called real because they do not include imaginary numbers and can be plotted on a real number line. Therefore, whenever you encounter a quadratic equation, applying the quadratic formula and evaluating the discriminant will guide you to finding its real solutions effectively.