Problem 62 - Algebra and Trigonometry

Question

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-6 x+10=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3 + i\) and \(x = 3 - i\).

Step 1: Identify the Quadratic Equation Structure

We are given the quadratic equation \(x^2 - 6x + 10 = 0\). It is in the standard form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -6\), and \(c = 10\).

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). Substituting the values from our equation, we get \(\Delta = (-6)^2 - 4 \times 1 \times 10 = 36 - 40 = -4\).

Step 3: Use the Quadratic Formula

The solutions of the quadratic equation are given by the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substituting \(b = -6\), \(a = 1\), and \(\Delta = -4\), we have \(x = \frac{6 \pm \sqrt{-4}}{2}\).

Step 4: Simplify the Square Root of the Discriminant

Since \(\Delta = -4\) is negative, the roots involve complex numbers. We know \(\sqrt{-4} = 2i\), where \(i\) is the imaginary unit. Thus, the expression becomes \(x = \frac{6 \pm 2i}{2}\).

Step 5: Simplify the Expression Further

Simplify the solutions: \(x = \frac{6}{2} \pm \frac{2i}{2} = 3 \pm i\).

Step 6: Express Solutions in the Form \(a+bi\)

The solutions \(x = 3 + i\) and \(x = 3 - i\) are already in the desired form \(a + bi\), where \(a = 3\) and \(b = \pm 1\).

Key Concepts

Complex Numbers

In mathematics, complex numbers take the form of \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) represents the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Complex numbers are used to extend the concept of one-dimensional real numbers to the two-dimensional complex plane, thereby providing a broader understanding of solutions to equations, especially those that do not have real solutions.
The real part of a complex number is \(a\), and the imaginary part is \(b\). When the discriminant of a quadratic equation is negative, as in our example, the roots are complex, meaning they will include the imaginary unit \(i\).
Here's why they matter:

• They allow the solution of equations that don't have real solutions.
• They are essential in various engineering fields where sinusoidal and exponential functions are used.

When approaching quadratic equations that produce imaginary roots, expect to express the solutions in the form of complex numbers, \(a + bi\).

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It is given by:
\[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]
This formula provides solutions by substituting the coefficients \(a\), \(b\), and \(c\) from the equation, along with the discriminant \(\Delta\).
The quadratic formula is very useful because:

• It can solve any quadratic equation, regardless of whether the solutions are real or complex.
• It bypasses the need to factor the quadratic expression manually.

In our example, the quadratic equation was \(x^2 - 6x + 10 = 0\), and applying this formula allowed us to find that \(x = 3 \pm i\). Always ensure all parts of the formula are addressed, particularly the square root part, which determines the nature of the roots.

Discriminant

The discriminant \(\Delta\) is a component of the quadratic formula and is crucial in determining the nature of the solutions to a quadratic equation. It is calculated using the formula:
\[\Delta = b^2 - 4ac\]
The value of the discriminant reveals several key aspects of the roots:

• If \(\Delta > 0\), there are two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution (a repeated root).
• If \(\Delta < 0\), the solutions are complex and involve the imaginary unit \(i\).

In our problem, the discriminant was \(-4\), which is less than zero, indicating that our equation has complex numbers as solutions. Understanding the discriminant helps to prepare for what type of numbers will appear in your solution.

Standard Form of Quadratic Equation

A quadratic equation in its standard form is presented as:
\[ax^2 + bx + c = 0\]
where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The structure of this form is important because it provides a consistent template to identify coefficients needed for solving the equation using the quadratic formula.
Recognizing this standard form is crucial because:

• It helps in applying the quadratic formula accurately.
• Makes it easier to calculate the discriminant.

In our exercise, the equation \(x^2 - 6x + 10 = 0\) was already in standard form, simplifying the solution process. Identifying coefficients \(a = 1\), \(b = -6\), and \(c = 10\) quickly led us into calculating the discriminant and using the formula effectively.