Problem 61 - Precalculus Mathematics for Calculus

Question

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+2 x+5=0$$

Step-by-Step Solution

Answer

The solutions are \(x = -1 + 2i\) and \(x = -1 - 2i\).

Step 1: Identify coefficients

The quadratic equation given is \(x^2 + 2x + 5 = 0\). Here, the coefficients are: \(a = 1\), \(b = 2\), and \(c = 5\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula \(D = b^2 - 4ac\). Substitute \(a = 1\), \(b = 2\), and \(c = 5\) into the formula: \[ D = 2^2 - 4 \times 1 \times 5 = 4 - 20 = -16 \] The discriminant is \(-16\), which is negative, indicating that the solutions will be complex numbers.

Step 3: Use the quadratic formula

The quadratic formula for finding roots is given by \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Using the coefficients \(a = 1\), \(b = 2\), and \(D = -16\): \[ x = \frac{-2 \pm \sqrt{-16}}{2 \times 1} = \frac{-2 \pm 4i}{2} \]

Step 4: Simplify the expression

Simplify the expression obtained from the quadratic formula: \[ x = \frac{-2 \pm 4i}{2} \] Divide each term by 2: \[ x = -1 \pm 2i \] Here, \(x_1 = -1 + 2i\) and \(x_2 = -1 - 2i\).

Step 5: Express solutions in standard form

The solutions to the equation in the standard form \(a + bi\) are thus: \(x_1 = -1 + 2i\) \(x_2 = -1 - 2i\)

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable, usually represented in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Understanding quadratic equations is essential because they appear frequently in various areas of mathematics and science. Solving a quadratic equation means finding the values of \( x \) that make the equation true. These solutions are also called roots of the equation.
Quadratic equations can have different types of solutions depending on the coefficients and the square root term in the solution process. They may have two distinct real solutions, one real solution (repeated roots), or two complex solutions.
In the equation \( x^2 + 2x + 5 = 0 \), we aim to find the solutions, which can be real or complex, by further exploring the concepts of discriminant and quadratic formula.

Discriminant

The discriminant is a crucial concept when it comes to understanding the nature of the roots of a quadratic equation. It is represented by the symbol \( D \) and is calculated using the formula \( D = b^2 - 4ac \).
Depending on the value of the discriminant, the roots of the quadratic equation can be predicted:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), the equation has exactly one real solution, known as a repeated or double root.
• If \( D < 0 \), the equation has two complex solutions.

In the given equation \( x^2 + 2x + 5 = 0 \), the discriminant is calculated as \( 4 - 20 = -16 \), which is negative. This indicates that the roots of this equation will indeed be complex numbers, as is confirmed by further calculations using the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of any quadratic equation. It is expressed as \( x = \frac{-b \pm \sqrt{D}}{2a} \), where \( D \) is the discriminant, and \( a \) and \( b \) are coefficients from the quadratic equation.
This formula provides a straightforward method to find the roots without factoring, making it especially useful for equations that do not factor easily.
Applying the quadratic formula to our equation \( x^2 + 2x + 5 = 0 \) with \( a = 1 \), \( b = 2 \), and \( D = -16 \), we calculate:

• \( x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} \)
• Simplifying this gives \( x = -1 \pm 2i \)

In this case, using the quadratic formula confirms the presence of complex solutions, which arise due to the negative discriminant.

Complex Solutions

Complex solutions occur when the discriminant \( D \) of a quadratic equation is less than zero. This means the solutions are not real numbers but rather complex numbers, which include an imaginary component. A complex number is written in the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit satisfying \( i^2 = -1 \).
The resulting solutions from our example equation \( x^2 + 2x + 5 = 0 \) are \( -1 + 2i \) and \( -1 - 2i \).
These solutions indicate that the roots of the equation do not intersect the x-axis, which is a key visual indicator of complex roots when graphing quadratic functions.

• Complex solutions often arise in real-world applications involving oscillatory systems, electrical engineering (AC circuits), and more.
• Understanding complex solutions is essential for those studying advanced mathematical topics and applied sciences.

Thus, in cases where real solutions are not found, complex numbers provide a comprehensive way to describe the roots of quadratic equations.