Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Find the solutions of the equation $$ x^{2}-2 x+26=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 1 + 5i \) and \( x = 1 - 5i \).

Step 1: Identify the Quadratic Equation

The quadratic equation given is \[ x^2 - 2x + 26 = 0 \].We need to solve this equation to find the values of \( x \).

Step 2: Determine the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is calculated using the formula\[ \Delta = b^2 - 4ac \].For the equation \( x^2 - 2x + 26 = 0 \), \( a = 1 \), \( b = -2 \), and \( c = 26 \).Thus, calculate \( \Delta = (-2)^2 - 4 \times 1 \times 26 = 4 - 104 = -100 \).

Step 3: Analyze the Discriminant

The discriminant \( \Delta = -100 \) is negative, indicating that the roots are complex and involve imaginary numbers.

Step 4: Use the Quadratic Formula to Find Solutions

The general formula to solve \( ax^2 + bx + c = 0 \) is \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \].Substitute \( a = 1 \), \( b = -2 \), and \( \Delta = -100 \) into the formula:\[ x = \frac{-(-2) \pm \sqrt{-100}}{2 \times 1} = \frac{2 \pm \sqrt{-100}}{2} \].The square root of \(-100\) is \( 10i \), where \( i \) is the imaginary unit \( i = \sqrt{-1} \). Thus, the solutions become:\[ x = \frac{2 \pm 10i}{2} \].

Step 5: Simplify the Solutions

Simplify \( x = \frac{2 \pm 10i}{2} \) to find:\[ x = 1 \pm 5i \].

Step 6: State the Solutions

The solutions to the quadratic equation \( x^2 - 2x + 26 = 0 \) are \( x = 1 + 5i \) and \( x = 1 - 5i \).

Key Concepts

Quadratic Formula

When we want to solve a quadratic equation, like the one given in the problem, we often use the quadratic formula. This formula is quite handy as it provides a direct way to find the roots of any quadratic equation.
To recall, a quadratic equation generally looks like this:
\[ ax^2 + bx + c = 0 \]
The quadratic formula to calculate the roots is as follows:

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

This formula uses the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
Here, the symbol \(\pm\) means there are two solutions: one with the plus sign and one with the minus.
The expression under the square root, \(b^2 - 4ac\), is called the discriminant. It will tell us the nature of the roots.

Discriminant Analysis

The discriminant is an essential part of the quadratic formula. It is specifically the portion \(b^2 - 4ac\) found under the square root.
This little expression packs quite a punch because it helps determine the type of roots you will find:

• If the discriminant is positive, you get two real roots.
• If it's zero, there is one real root (also known as a repeated root).
• But if it's negative, you get complex roots involving imaginary numbers.

In the example provided, we found that:
\(\Delta = -100\)
This negative discriminant indicates that the solutions are going to involve complex numbers.
Hence, they won't be found on the number line like real numbers.

Imaginary Unit

Complex numbers use the imaginary unit, often denoted as \(i\). This is defined as follows:

• \(i = \sqrt{-1}\)

The imaginary unit comes into play when we deal with a negative discriminant, as seen in the quadratic equation we solved.
When taking the square root of a negative number, such as \(-100\), you express it using \(i\):

• \(\sqrt{-100} = 10i\)

Using \(i\) allows us to express solutions to equations even when our discriminant is negative.
In this problem, the solutions became \(x = 1 \pm 5i\), meaning the answers are neither purely real nor all imaginary, but a combination of both, which we call complex numbers.