Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(39-56:\) Find the solutions of the equation. $$ x^{2}+4 x+13=0 $$

Step-by-Step Solution

Answer

The solutions are complex: \(x = -2 + 3i\) and \(x = -2 - 3i\).

Step 1: Identify the Equation Form

We begin by noting that the given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 4\), and \(c = 13\). Our goal is to find the values of \(x\) that satisfy this equation.

Step 2: Apply the Quadratic Formula

The quadratic formula, \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\), allows us to find the roots of a quadratic equation. Applying this formula, we substitute \(a = 1\), \(b = 4\), and \(c = 13\).

Step 3: Calculate the Discriminant

The discriminant \(D = b^2 - 4ac\) helps to determine the nature of the roots. Calculate \(D = 4^2 - 4 \cdot 1 \cdot 13 = 16 - 52 = -36\). Since \(D < 0\), the roots are complex (not real).

Step 4: Find the Complex Roots Using the Quadratic Formula

Since the discriminant is negative, the roots are complex. Substituting into the quadratic formula, we have:\[x = \frac{{-4 \pm \sqrt{-36}}}{2 \cdot 1}\]Simplify it to:\[x = \frac{{-4 \pm i\sqrt{36}}}{2} = \frac{{-4 \pm 6i}}{2}\]Therefore, the solutions are \(x = -2 + 3i\) and \(x = -2 - 3i\).

Key Concepts

Complex Roots

When you encounter a quadratic equation with a negative discriminant, the solutions are not real numbers, but complex numbers. Complex roots arise because you must take the square root of a negative number.
In mathematics, this introduces the imaginary unit, denoted by \(i\), where \(i = \sqrt{-1}\). Therefore, a square root of any negative number can be expressed in terms of \(i\).
So, for the equation \(x^2 + 4x + 13 = 0\), the roots are complex because the discriminant \(b^2 - 4ac\) is negative. Calculating gives \(-36\), leading to complex roots:

• \(x = -2 + 3i\)
• \(x = -2 - 3i\)

This indicates that these roots are conjugate pairs, a common feature when solving quadratics with real coefficients.Don't be intimidated by complex numbers! They often appear in physics and engineering, and represent phenomena that real numbers cannot describe alone.

Quadratic Formula

The quadratic formula is a powerful tool that helps us find solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is expressed as:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}.\]Using this formula, you can determine the roots of a quadratic equation by substituting in the values for \(a\), \(b\), and \(c\).

This formula consists of two key elements: the portion \(-b/2a\), which gives the central axis of the parabola described by the equation, and the square root part, \(\sqrt{b^2 - 4ac}/2a\), which determines the nature and distance of the roots from this axis.

The quadratic formula is considered universal because it applies to any quadratic equation, regardless of its complexity. Whether you encounter real or complex solutions, this formula will help find the roots efficiently and effectively.

Discriminant

The discriminant is a crucial component in determining the nature of the roots of a quadratic equation. It is found inside the quadratic formula, specifically as the expression \(b^2 - 4ac\).

Here's what the discriminant tells us:

• If \(D = 0\), there is one real and repeated root.
• If \(D > 0\), there are two distinct real roots.
• If \(D < 0\), as seen in our example, the roots are complex and conjugate.

In the equation \(x^2 + 4x + 13 = 0\), we calculated \(D = -36\). Since \(D < 0\), we recognize that the roots are complex. This guides us to use the quadratic formula with care to uncover these complex solutions.

By examining the discriminant before attempting to solve the equation, one can anticipate whether the solutions will be real or complex, and approach the problem more strategically.