Problem 102 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3} + \frac{\sqrt{2}}{3}i\) and \(x = \frac{1}{3} - \frac{\sqrt{2}}{3}i\).

Step 1: Identify the quadratic formula

To solve a quadratic equation of the form \(ax^2 + bx + c = 0\), we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. For this equation, \(a=3\), \(b=-2\), and \(c=1\).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation is given by \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula to get \((-2)^2 - 4 imes 3 imes 1 = 4 - 12 = -8\).

Step 3: Evaluate the square root of the discriminant

Since the discriminant is negative, \(-8\), we need to calculate the square root in terms of complex numbers. \(\sqrt{-8}\) can be expressed as \(\sqrt{8}i\). Simplifying \(\sqrt{8}\) yields \(2\sqrt{2}\), so \(\sqrt{-8} = 2\sqrt{2}i\).

Step 4: Use the quadratic formula

Substitute the values \(a=3\), \(b=-2\), and \(\sqrt{-8} = 2\sqrt{2}i\) into the quadratic formula: \[x = \frac{-(-2) \pm 2\sqrt{2}i}{2 \times 3} = \frac{2 \pm 2\sqrt{2}i}{6}\].

Step 5: Simplify the expression

Simplify the expression \(\frac{2 \pm 2\sqrt{2}i}{6}\) by dividing each term by 6: \[x = \frac{1}{3} \pm \frac{\sqrt{2}}{3}i\].

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are crucial for solving equations that do not have real solutions. They consist of two parts: a real part and an imaginary part. A complex number is usually written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The \(i\) in \(bi\) stands for the imaginary unit, defined as \(i = \sqrt{-1}\).
Understanding complex numbers is essential for solving quadratic equations with negative discriminants. Negative discriminants indicate that the solutions are not real numbers but complex. When solving such equations, the imaginary number \(i\) helps express the square roots of negative numbers.

• For example, \( \sqrt{-8} \) is expressed as \( 2\sqrt{2}i \), signifying a complex solution.
• This expression helps solve quadratic equations where the standard algebraic solutions don't apply, making complex numbers indispensable tools in mathematics.

Discriminant

The discriminant of a quadratic equation plays a critical role in determining the nature of its solutions. It's calculated from the quadratic equation \(ax^2 + bx + c = 0\) using the formula \(b^2 - 4ac\).
There are three possible scenarios concerning the discriminant:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, the equation has exactly one real root, also known as a repeated root.
• When the discriminant is negative, as is the case in this example with \(-8\), it indicates that the equation has two complex roots.

A negative discriminant requires using complex numbers to find the roots of the equation. This connection between the discriminant and the types of roots it predicts is fundamental in understanding and solving quadratic equations effectively.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation \(ax^2 + bx + c = 0\). It is expressed as:

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

This formula is derived from completing the square method and provides a structured way to solve quadratics by plugging in the coefficients of the equation.

How to use the Quadratic Formula

• First, identify and substitute the values of \(a\), \(b\), and \(c\) from the equation into the formula.
• Next, compute the discriminant \(b^2 - 4ac\) and evaluate its square root.
• Finally, solve for \(x\) using the "+" and "-" options to account for both potential solutions.

Using the quadratic formula is advantageous because it works universally for all kinds of quadratic equations, regardless of whether the solutions are real or complex. In situations where the discriminant is negative, as in the sample equation, the quadratic formula smoothly adapts to deliver the complex roots, showcasing its versatility and power.