Problem 49 - Elementary Algebra

Question

\(z^{2}-20 z+164=0\)

Step-by-Step Solution

Answer

The solutions are \(z = 10 + 8i\) and \(z = 10 - 8i\).

Step 1 - Identify coefficients

The given equation is a quadratic equation in the form of \(ax^2 + bx + c = 0\). Identify the coefficients: \(a = 1\), \(b = -20\), and \(c = 164\).

Step 2 - Compute the discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substitute the values to find the discriminant: \(\Delta = (-20)^2 - 4 \cdot 1 \cdot 164 = 400 - 656 = -256\).

Step 3 - Analyze the discriminant

Since the discriminant \(\Delta = -256\) is less than zero, the quadratic equation has two complex solutions.

Step 4 - Use the quadratic formula

The quadratic formula is given by \[z = \frac{-b \pm \sqrt{\Delta}}{2a}\]. Substituting the values, we get \[z = \frac{-(-20) \pm \sqrt{-256}}{2 \cdot 1}\].

Step 5 - Simplify the expression

Simplify the expression under the square root and perform the division:\[z = \frac{20 \pm \sqrt{256i^2}}{2}\], where \(i\) is the imaginary unit. Therefore, \(\sqrt{-256} = 16i\).

Step 6 - Calculate the roots

Continue simplifying to get the roots:\[z = \frac{20 \pm 16i}{2}\]\[z = 10 \pm 8i\].

Key Concepts

quadratic formula

To solve quadratic equations, we often use the quadratic formula. This formula provides a way to find the roots of any quadratic equation of the form h4.

Let's breakdown the quadratic formula step by step:

• The term
• The fraction
• The part under the square root

Using the quadratic formula, we can solve the given equation

Using the quadratic formula, you can confidently solve any quadratic equation!

discriminant

The discriminant provides useful information about the nature of the roots. It is found inside the quadratic formula in the part under the square root sign. You calculate it using the formula

Let's go through this step-by-step:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (also known as a repeated root).
• If the discriminant is negative, there are two complex solutions.

In this exercise, we found the discriminant

This negative discriminant tells us that our solutions will be complex.

complex solutions

When the discriminant is negative, our quadratic equation has complex solutions. Complex solutions come in conjugate pairs. They have the form

In practice, this means one solution will be

In our example, the complex solutions are

These numbers cannot be placed on the regular number line; instead, they can be shown on a complex plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

imaginary unit

The imaginary unit is denoted by

The idea behind the imaginary unit is to extend the real number system. Sometimes, we need to calculate the square root of a negative number.

To understand the imaginary unit better, remember these fundamental rules:

In solving the quadratic equation from our exercise, we encountered :

This use of the imaginary unit allowed us to find the complex solutions.