Problem 105 - Calculus for Biology and Medicine

Question

Solve each quadratic equation in the complex number system. \(4 x^{2}-3 x+1=0\)

Step-by-Step Solution

Answer

The roots of the equation are \(x = \frac{3 + i\sqrt{7}}{8}\) and \(x = \frac{3 - i\sqrt{7}}{8}\).

Step 1: Identify the coefficients

The given quadratic equation is \(4x^2 - 3x + 1 = 0\). The coefficients can be identified as follows: \(a = 4\), \(b = -3\), and \(c = 1\).

Step 2: Use the quadratic formula

To find the roots of the quadratic equation, we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the discriminant

The discriminant \(\Delta\) is calculated using \(b^2 - 4ac\). Substitute the values to get \(\Delta = (-3)^2 - 4 \cdot 4 \cdot 1 = 9 - 16 = -7\).

Step 4: Determine the nature of the roots

Since the discriminant is negative (\(\Delta = -7\)), the quadratic equation has two complex roots.

Step 5: Calculate the complex roots

Substitute the values into the quadratic formula to find the roots: \[x = \frac{-(-3) \pm \sqrt{-7}}{2 \cdot 4}\] \[x = \frac{3 \pm \sqrt{-7}}{8}\] Express the roots in terms of imaginary numbers: \(x = \frac{3 \pm i\sqrt{7}}{8}\).

Key Concepts

Complex Numbers

Complex numbers allow us to expand our understanding of numbers beyond the real number line. They consist of a real part and an imaginary part and are often expressed in the form of \(a + bi\). Here, \(a\) is the real component, while \(bi\) represents the imaginary component, with \(i\) being the square root of \(-1\). This imaginary unit \(i\) is crucial because it lets us define numbers that have no real square roots, such as negative numbers.
For example, \(\sqrt{-7}\) can be expressed as \(i\sqrt{7}\), using the imaginary unit. Thus, complex numbers become essential when dealing with quadratic equations that yield a negative discriminant, as they allow us to express non-real roots in a meaningful way.

Discriminant

The discriminant is a key component of the quadratic formula, used to determine the nature of the roots of a quadratic equation. It is represented by \( \Delta \), and is calculated as \( b^2 - 4ac \), where \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
The value of the discriminant provides important insights into the type and number of roots:

• If \( \Delta > 0\), the equation has two distinct real roots.
• If \( \Delta = 0\), there is exactly one real root, also called a repeated root.
• If \( \Delta < 0\), the equation has two complex roots, as shown in the original exercise, where \( \Delta = -7 \).

Thus, the discriminant not only helps determine if solutions are real or complex but also guides us in approaching different solution strategies.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula provides the solutions to the quadratic equation \(ax^2 + bx + c = 0\). Here’s how each part plays a crucial role:

• \(-b\): The negation of the linear coefficient, which helps shift the roots correctly on the number line.
• \(\pm \sqrt{b^2 - 4ac}\): This term is central as it involves the discriminant \(\Delta\), determining the nature and type of roots.
• \(2a\): This division accounts for the scaling effect of the quadratic coefficient to ensure the correct position of the roots.

In the example of the original exercise, after calculating \( \Delta = -7\), we find complex roots rather than real, and they can be expressed using the quadratic formula, leading to roots like \( \frac{3 \pm i\sqrt{7}}{8} \). This illustrates the formula's versatility, allowing it to handle all types of roots efficiently, whether real or complex.