Problem 101 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{3 \pm i \sqrt{7}}{4}\).

Step 1: Identify the coefficients

The quadratic equation given is \(2x^2 - 3x + 2 = 0\). The coefficients can be identified as \(a = 2\), \(b = -3\), and \(c = 2\). This set of coefficients will be used in the quadratic formula to find the roots.

Step 2: Write the quadratic formula

The quadratic formula, used to solve equations of the form \(ax^2 + bx + c = 0\), is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This will be used to find the solutions of the equation.

Step 3: Calculate the discriminant

The discriminant of a quadratic equation is given by \(b^2 - 4ac\). Substituting the known values gives \((-3)^2 - 4 \cdot 2 \cdot 2 = 9 - 16 = -7\). A negative discriminant indicates that the roots will be complex numbers.

Step 4: Apply the quadratic formula

Substitute the coefficients and the discriminant into the quadratic formula: \[x = \frac{-(-3) \pm \sqrt{-7}}{2 \cdot 2}\] which simplifies to \[x = \frac{3 \pm \sqrt{-7}}{4}\].

Step 5: Solve for the complex roots

Since the discriminant is negative, \(\sqrt{-7}\) can be expressed as \(i \sqrt{7}\) where \(i\) is the imaginary unit. Thus, the solutions are \[x = \frac{3 \pm i \sqrt{7}}{4}\]. These are the complex roots of the equation.

Key Concepts

Quadratic Equation

A quadratic equation is an essential concept in algebra. It's an equation where the highest exponent of the variable is 2, typically written in the form:

• \( ax^2 + bx + c = 0 \)

where:

• \(a\), \(b\), and \(c\) are constants with \(a eq 0\)

In our example, the quadratic equation given is \(2x^2 - 3x + 2 = 0\). Quadratic equations can have different types of solutions:

• Two distinct real solutions
• One real solution (a double root)
• Two complex solutions

When solving quadratic equations, the nature of the solutions depends heavily on a component called the discriminant. We'll dive into that next.

Discriminant

The discriminant is a key concept in understanding the nature of the roots of a quadratic equation. It is calculated from the coefficients of the quadratic equation. Specifically, the formula for the discriminant \(D\) is:

• \( D = b^2 - 4ac \)

This value reveals vital information about the solutions:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, a double root.
• If \(D < 0\), the solutions are complex numbers.

In our problem, the discriminant is calculated as \((-3)^2 - 4 \cdot 2 \cdot 2 = -7\). The negative discriminant \(-7\) indicates that the equation has complex roots.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It provides a straightforward way to solve \( ax^2 + bx + c = 0 \) by plugging in the coefficients. This formula is:

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

Using the quadratic formula involves these steps:

• Identify and substitute the values of \(a\), \(b\), and \(c\).
• Calculate the discriminant \(b^2 - 4ac\).
• Substitute back into the formula to find the roots.

In our exercise, substituting the coefficients into the formula gives:\[x = \frac{-(-3) \pm \sqrt{-7}}{4} = \frac{3 \pm i \sqrt{7}}{4}\]. Thus, we obtain complex roots as \(i\), the imaginary unit, is employed to represent the square root of negative numbers.