Problem 18 - Elementary Algebra

Question

Choose the appropriate method to solve the following. $$ 3 x 2-x+2=0 $$

Step-by-Step Solution

Answer

There are no real solutions; the equation has complex roots.

Step 1: Understanding the Equation

First, look at the equation given: \(3x^2 - x + 2 = 0\). This is a quadratic equation because the highest power of \(x\) is 2. Quadratic equations are generally of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants.

Step 2: Applying the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In our equation, \(a = 3\), \(b = -1\), and \(c = 2\). These values will be substituted into the formula to solve for \(x\).

Step 3: Calculating the Discriminant

Calculate the discriminant, \(b^2 - 4ac\). Substitute \(b = -1\), \(a = 3\), and \(c = 2\) to get: \((-1)^2 - 4 \cdot 3 \cdot 2 = 1 - 24 = -23\).

Step 4: Analyzing the Discriminant

The discriminant \(-23\) is less than zero, which means there are no real solutions to the equation. The solutions will be complex numbers.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve equations of the form \( ax^2 + bx + c = 0 \). It gives us a systematic way to find the solutions (or roots) of quadratic equations. The formula is expressed as:

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

Here, \( a \), \( b \), and \( c \) are coefficients from the equation. This formula can be used to find both real and complex solutions, depending on the discriminant.

The term \( \pm \) indicates that there are typically two solutions since the square root can have both positive and negative values. Always substitute the equation's coefficients carefully to avoid mistakes in calculations. Understanding the quadratic formula is essential for dealing with any quadratic equation you encounter.

Discriminant

The discriminant is a magical component of the quadratic formula. It is the part under the square root: \( b^2 - 4ac \). By calculating the discriminant, we can predict the nature of the roots without even solving the equation completely.

• If the discriminant is positive (\( > 0 \)), the quadratic equation has two real and distinct solutions.
• If it is zero (\( = 0 \)), there is one real solution, as both roots are equal.
• If it is negative (\( < 0 \)), the solutions are complex or imaginary numbers.

Knowing the value of the discriminant saves time by giving you insight into what to expect from the solutions. This understanding is particularly useful when solving multiple quadratic equations efficiently.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is negative. This means there are no real solutions, but instead, the solutions involve imaginary numbers.

In mathematics, imaginary numbers are used to extend the real number system and are based on the imaginary unit \( i \), where \( i = \sqrt{-1} \). In such cases, the solutions to the quadratic equation are:

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

Here, \( |b^2 - 4ac| \) represents the absolute value of the discriminant, effectively turning it positive for calculation purposes.

Complex solutions are a significant part of advanced mathematics, showing up in various fields, from engineering to computer graphics. Understanding these concepts is essential for anyone delving deeper into mathematics and its applications.