Problem 106 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = 1 + \frac{\sqrt{2}}{2}i\) and \(x = 1 - \frac{\sqrt{2}}{2}i\).

Step 1: Identify the Coefficients

The given quadratic equation is \(-2x^2 + 4x - 3 = 0\). We identify the coefficients as \(a = -2\), \(b = 4\), and \(c = -3\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Substitute the values to get \(4^2 - 4(-2)(-3) = 16 - 24 = -8\). Since it's negative, the solutions will be complex.

Step 3: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). We substitute \(a = -2\), \(b = 4\), and the calculated discriminant \(-8\) into the formula: \(x = \frac{-4 \pm \sqrt{-8}}{-4}\).

Step 4: Simplify the Square Root

Since the discriminant is negative, we simplify \(\sqrt{-8}\) as \(\sqrt{8}i\). Note that \(\sqrt{8} = 2\sqrt{2}\), so \(\sqrt{-8} = 2\sqrt{2}i\).

Step 5: Simplify the Expression

Substitute \(\sqrt{-8} = 2\sqrt{2}i\) back into the quadratic formula: \(x = \frac{-4 \pm 2\sqrt{2}i}{-4}\). This can be further simplified, giving \(x = 1 \mp \frac{\sqrt{2}}{2}i\), resulting in \(x_1 = 1 + \frac{\sqrt{2}}{2}i\) and \(x_2 = 1 - \frac{\sqrt{2}}{2}i\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree 2. It is generally given in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable to be solved. The importance of quadratic equations lies in their ability to model various real-world phenomena ranging from physics to finance.
Quadratic equations can have different types of solutions:

• Two distinct real solutions
• One real solution (repeated)
• Two complex solutions

In the case of our original exercise, the equation \(-2x^2 + 4x - 3 = 0\) is a quadratic equation. Here, the values of the coefficients are \(a = -2\), \(b = 4\), and \(c = -3\). The type of solutions we get depends on the discriminant value which is discussed in the next section.

Discriminant

The discriminant is a key component in determining the nature of the solutions of a quadratic equation. It is found using the formula \(b^2 - 4ac\). Depending on its value, we can predict the nature of the roots of the quadratic equation. Here's what different discriminant values tell us:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If it is negative, the roots are complex and come as conjugate pairs.

In our exercise, we calculated the discriminant as \(4^2 - 4(-2)(-3) = 16 - 24 = -8\). Because this result is negative, it indicates that the solutions will be complex numbers. We will cover how to solve these using the quadratic formula in the next section.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is given by the formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula allows us to solve quadratic equations even when other methods like factoring are not possible. In our example, after substituting the values of \(a = -2\), \(b = 4\), and the discriminant \(-8\), into the quadratic formula, we get:
\[x = \frac{-4 \pm \sqrt{-8}}{-4}\]
Since the discriminant is negative, we adjust the square root to involve complex numbers. Simplifying \(\sqrt{-8}\) gives \(2\sqrt{2}i\). This is plugged back into the formula, resulting in:
\[x = 1 \mp \frac{\sqrt{2}}{2}i\]
Thus, the solutions of our equation are two complex numbers: \(x_1 = 1 + \frac{\sqrt{2}}{2}i\) and \(x_2 = 1 - \frac{\sqrt{2}}{2}i\). These solutions are complex conjugates of each other, typical whenever the discriminant is negative.