Problem 102 - Calculus for Biology and Medicine

Question

Solve each quadratic equation in the complex number system. \(x^{2}+2 x+3=0\)

Step-by-Step Solution

Answer

The roots of the equation are \(-1 + i \sqrt{2}\) and \(-1 - i \sqrt{2}\).

Step 1: Identify a, b, and c

In the quadratic equation \[x^2 + 2x + 3 = 0\] the coefficients are: \(a = 1\), \(b = 2\), and \(c = 3\). Identify these values for use in the quadratic formula.

Step 2: Write the Quadratic Formula

The quadratic formula used to find the roots of \(ax^2 + bx + c = 0\) is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\), to check if the roots are real or complex.\[b^2 - 4ac = 2^2 - 4 \times 1 \times 3 = 4 - 12 = -8\].Since the discriminant is negative, the roots are complex numbers.

Step 4: Substitute into the Quadratic Formula

Substitute \(a\), \(b\), and the discriminant into the quadratic formula:\[x = \frac{-2 \pm \sqrt{-8}}{2 \times 1}\].

Step 5: Simplify the Expression

Simplify the expression inside the square root and solve:\[\sqrt{-8} = \sqrt{-1 \times 8} = \sqrt{-1} \times \sqrt{8} = i \cdot \sqrt{8} = 2i \sqrt{2}\].This leads to:\[x = \frac{-2 \pm 2i \sqrt{2}}{2}\].

Step 6: Simplify the Roots

Divide each term in the numerator by 2:\[x = -1 \pm i \sqrt{2}\].Thus, the solutions are \(-1 + i \sqrt{2}\) and \(-1 - i \sqrt{2}\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real number system. They include all numbers in the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined by \(i^2 = -1\). This means complex numbers encompass both a real and an imaginary part, which allows us to express solutions that aren't possible with real numbers alone.

• Real Part: The \(a\) in \(a + bi\). Can be any real number, including zero.
• Imaginary Part: The \(bi\) in \(a + bi\). Represents multiples of the imaginary unit.

In the context of quadratic equations, when the discriminant (the part of the quadratic after the minus sign under the square root) is negative, the solutions are complex numbers. This means we use the imaginary unit \(i\) when calculating these roots to account for the square root of negative numbers.

Quadratic Formula

The quadratic formula is a universal tool used to find the roots of quadratic equations of the form \(ax^2 + bx + c = 0\). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is powerful because it provides a direct way to solve any quadratic equation, no matter if the solutions are real or complex.Here are the components of the formula:

• \(-b\): Negates the coefficient \(b\).
• \(\pm\): Indicates two possible solutions due to the square root operation.
• \(\sqrt{b^2 - 4ac}\): The discriminant, determines the nature of the roots.
• \(2a\): The denominator, which normalizes the entire expression.

Using this formula, you can confidently solve quadratic equations without needing to venture into guessing or specific factoring methods.

Discriminant

The discriminant is a key part of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. It is calculated as:\[b^2 - 4ac\].The value of the discriminant reveals the type of solutions:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, sometimes known as a repeated root.
• If the discriminant is negative, as seen in our problem, the equation has two complex roots.

When the discriminant is negative, the square root of a negative number arises, leading us to employ complex numbers (notably \(i\), the imaginary unit) to express the roots.

Roots of Equations

Finding the roots of a quadratic equation means solving for the values of \(x\) that make the equation true. These values are often referred to as solutions or zeros of the equation.In our specific equation \(x^2 + 2x + 3 = 0\), we've determined that the roots are complex numbers. By employing the quadratic formula and considering the negative discriminant, we derived the roots:

• \(-1 + i \sqrt{2}\)
• \(-1 - i \sqrt{2}\)

These roots indicate that the parabola represented by the original quadratic equation does not intersect the \(x\)-axis, as real roots would. Instead, it aligns entirely within the realm of complex numbers, exhibiting no real-number solutions. Understanding this concept is critical as it demonstrates the flexibility and depth of algebraic solutions beyond real-world constraints.