Problem 104 - 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 roots are \(x = 1 + \frac{i\sqrt{2}}{2}\) and \(x = 1 - \frac{i\sqrt{2}}{2}\).

Step 1: Identify the Coefficients

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

Step 2: Calculate the Discriminant

The discriminant \(D\) is calculated using the formula \(D = b^2 - 4ac\). Substitute the values: \(D = 4^2 - 4(-2)(-3) = 16 - 24 = -8\).

Step 3: Determine the Nature of the Roots

Since the discriminant \(D = -8\) is less than zero, the equation has two complex roots.

Step 4: Calculate the Complex Roots using the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots. Substitute the values: \(x = \frac{-4 \pm \sqrt{-8}}{-4}\).

Step 5: Simplify the Roots

Simplify the expression \(x = \frac{-4 \pm 2i\sqrt{2}}{-4}\). This simplifies to \(x = 1 \pm \frac{i\sqrt{2}}{2}\). Therefore, the roots are \(x = 1 + \frac{i\sqrt{2}}{2}\) and \(x = 1 - \frac{i\sqrt{2}}{2}\).

Key Concepts

Understanding Complex Numbers

In mathematics, especially in the realm of quadratic equations, complex numbers often make an appearance. When an equation doesn't have real solutions, it has what we call complex roots. A complex number is of the form \(a + bi\), where:

• \(a\) is the real part,
• and \(bi\) is the imaginary part, which involves the imaginary unit \(i\).

The imaginary unit \(i\) is defined such that \(i^2 = -1\). This fascinating concept helps us deal with the square roots of negative numbers, which don't exist in the realm of real numbers.
In our exercise, the discriminant was negative (\(D = -8\)), leading us to a complex solution. Anytime the discriminant is less than zero, expect to see complex numbers involved. The presence of complex numbers allows for a comprehensive solution when the parabola doesn't intersect the x-axis. So, next time you encounter a negative square root in solving quadratics, think of it as an entry into the world of complex numbers.

Recognizing the Discriminant

The discriminant is a valuable tool in determining the nature of the roots of a quadratic equation. It is extracted from the quadratic formula and given by the expression \(b^2 - 4ac\). Here's what the discriminant, \(D\), can tell you:

• \(D > 0\): two distinct real roots
• \(D = 0\): one real root (also called a double root)
• \(D < 0\): two complex roots

In simpler terms, the discriminant helps classify what kind of solutions you should expect.
For the equation \(-2x^2 + 4x - 3 = 0\), calculating the discriminant results in \(D = -8\), which immediately signals complex roots. This means our solution won’t be on the real number line. Understanding the discriminant saves time and sets your expectations about the kind of answer you'll derive.

Applying the Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation. It is represented as \(x = \frac{-b \pm \sqrt{D}}{2a}\). Here’s how it works:

• \(b\) is the coefficient of \(x\)
• \(a\) is the coefficient of \(x^2\)
• \(c\) is the constant term
• \(\pm\) indicates two potential solutions: one plus and one minus

Once you've identified the coefficients and calculated the discriminant \(D\), it's time to substitute them into the quadratic formula.
For our exercise, when we plugged values into \(x = \frac{-4 \pm \sqrt{-8}}{-4}\), we simplified it using \(i\) (since \(\sqrt{-1} = i\)) and arrived at \(x = 1 \pm \frac{i\sqrt{2}}{2}\). This neatly gives us two complex solutions.
Using the quadratic formula thus transforms the intimidating process of finding roots into a straightforward plug-and-play method. It’s crucial to understand this formula well as it is a cornerstone of solving quadratic equations.