Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve \(\frac{1}{3} x^{2}-2 x+9=0\)

Short answer

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

Step-by-step solution

Identify the coefficients

Write down the general form of a quadratic equation, which is \[ ax^2 + bx + c = 0 \]. For the given equation \( \frac{1}{3}x^2 - 2x + 9 = 0 \), identify the coefficients: \( a = \frac{1}{3} \), \( b = -2 \), and \( c = 9 \).

Substitute the coefficients into the quadratic formula

The quadratic formula is \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Plugging in the values \( a = \frac{1}{3} \), \( b = -2 \), and \( c = 9 \) into the formula, we get \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \left( \frac{1}{3} \right)(9)}}{2 \left( \frac{1}{3} \right)} \].

Simplify the expressions within the quadratic formula

Calculate the discriminant: \[ (-2)^2 - 4 \left( \frac{1}{3} \right)(9) = 4 - 12 = -8 \]. Since the discriminant is negative, \( \sqrt{-8} \) involves imaginary numbers. Simplify further: \[ x = \frac{2 \pm \sqrt{4 (-2)}}{ \frac{2}{3}} = \frac{2 \pm 2i\sqrt{2}}{ \frac{2}{3}} \].

Simplify the fraction

To simplify, multiply numerator and denominator by 3 to get: \[ x = \frac{6 \pm 6i\sqrt{2}}{2} = 3 \pm 3i\sqrt{2} \].

Key concepts

Understanding the Quadratic Formula

Let's start by exploring the quadratic formula, a key tool for solving quadratic equations. Quadratic equations are standardly written as \[ ax^2 + bx + c = 0 \]. The quadratic formula to solve these equations is given by \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Here's what the components mean:

• \( a \) represents the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

This formula provides solutions for \( x \), which are the roots of the quadratic equation. These roots can be real or imaginary, based on the value of the discriminant, which we'll delve into shortly.

It’s important to memorize the quadratic formula as it’s a fundamental solution method for quadratic equations. Practice it by solving different quadratic equations until you feel comfortable with using it.

The Role of Imaginary Numbers

Imaginary numbers come into play when dealing with the square roots of negative numbers. The basic imaginary unit \( i \) is defined as \( \sqrt{-1} \). When the discriminant of your quadratic equation is negative, your solutions will include imaginary numbers.

For example, in our problem, the discriminant turned out to be \(-8\). Taking the square root of \(-8\) introduces \( i \), because \( \sqrt{-8} = \sqrt{4 \cdot (-2)} = 2i\sqrt{2} \). This makes the quadratic formula solution include imaginary numbers: \[ x = \frac{2 \pm 2i\sqrt{2}}{ \frac{2}{3}} \].

Don’t be intimidated by imaginary numbers. They are as valid as real numbers and play crucial roles in various fields such as engineering and physics. They allow for solutions to equations that would otherwise have no real solutions.

Understanding the Discriminant

The discriminant is a part of the quadratic formula found inside the square root: \[ b^2 - 4ac \]. The value of the discriminant tells us about the nature of the roots:

• If the discriminant > 0: Two distinct real roots
• If the discriminant = 0: One real root (repeated)
• If the discriminant < 0: Two complex roots (involving imaginary numbers)

In our example, the discriminant is calculated as follows: \[ (-2)^2 - 4 \left( \frac{1}{3} \right)(9) = 4 - 12 = -8 \]. Since the discriminant is negative, we know the quadratic equation has two complex roots. This is why we ended with \[ x = 3 \pm 3i\sqrt{2} \] as our final answer.

Understanding the discriminant helps you quickly determine the number and type of solutions without fully solving the quadratic equation. This insight can save time and clarify the nature of the equation’s roots.