Problem 89 - College Algebra

Question

Solve each equation in Exercises 73-98 by the method of your choice. \(x^{2}-6 x+13=0\)

Step-by-Step Solution

Answer

The roots of the quadratic equation \(x^{2}-6x+13=0\) are \(x = 3 + 2i\) and \(x = 3 - 2i\).

Step 1: Identify a, b, and c

Identify the coefficients from the equation \(x^{2}-6x+13=0\) and place them into the quadratic formula. Here, \(a=1\), \(b=-6\), and \(c=13\).

Step 2: Plug values into the quadratic formula

Substitute the values \(a=1\), \(b=-6\), and \(c=13\) into the quadratic formula. Obtain \(x = \frac{-(-6) \pm \sqrt{(-6)^{2}-4*1*13}}{2*1}\). This simplifies to: \(x = \frac{6 \pm \sqrt{36 - 52}}{2}\).

Step 3: Solve the square root

Calculate the value under the square root sign: \(36 - 52 = -16\). Since we cannot square root a negative number in the real number system, the roots of the equation will be complex numbers. Obtain \(x = \frac{6 \pm \sqrt{-16}}{2}\).

Step 4: Simplify the expression

Simplify the final expression to find the complex roots. We obtain from \(\sqrt{-16}=4i\) the roots \(x = 3 + 2i\) and \(x = 3 - 2i\).

Key Concepts

Complex Numbers

In mathematics, complex numbers are numbers that combine real and imaginary parts. They extend our number system beyond the real numbers to address problems such as taking the square root of a negative number, which cannot be done with real numbers alone. A complex number is expressed in the form of

• \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit.

The imaginary unit \( i \) is defined as the square root of -1. This means that \( i^2 = -1 \).
Complex numbers are crucial in solving certain equations, such as quadratic equations, that have no real solutions.Now, let’s revisit the problem at hand. The equation \( x^2 - 6x + 13 = 0 \) results in a negative value under the square root when solved using the quadratic formula. To express the solution, we need complex numbers.

• After simplifying, the expression under the square root reveals \( \sqrt{-16} \), which gives us the imaginary number \( 4i \).
• Thus, the solutions become \( 3 + 2i \) and \( 3 - 2i \), both of which are complex numbers.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations that are in the standard form:

• \( ax^2 + bx + c = 0 \).

This formula is universally applicable, offering a systematic way to find the equation's roots. The quadratic formula is given by:

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

However, interpreting the results requires understanding what the discriminant, \( b^2 - 4ac \), represents.

The Discriminant and Its Importance

The discriminant can tell us the nature of the roots:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root, a repeated one.
• If \( b^2 - 4ac < 0 \), the roots are complex, indicating imaginary numbers play a role in the solutions.

In our exercise, the discriminant \( 36 - 52 = -16 \) is negative, leading to the roots being complex numbers. Once the quadratic formula is applied, remember to adjust your expectation to the nature of the calculations. Seeing a negative under the square root shouldn't be unexpected; it's merely a door to exploring complex numbers.

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value(s) of unknown variables that satisfy the given equality. Different types of equations require different solving techniques, and one very common type is the quadratic equation, which typically has the form \( ax^2 + bx + c = 0 \).

Several methods can be employed to solve quadratic equations:

• Factoring, if the equation can be rewritten as a product of factors.
• Completing the square, which involves rewriting the equation in perfect square form.
• Using the quadratic formula, as shown in our problem.

Example Solution Process

In the original exercise, solving the equation \( x^2 - 6x + 13 = 0 \) started with identifying coefficients \( a \), \( b \), and \( c \). Here's a streamlined approach:

• First, recognize the equation as a quadratic.
• Use the quadratic formula, substitute \( a = 1 \), \( b = -6 \), and \( c = 13 \).
• Calculate the discriminant to determine the nature of the roots.
• Solve by following the operation under the square root to determine if the solution involves complex numbers.
• Simplify the expression to find the roots of the equation: \( x = 3 + 2i \) and \( x = 3 - 2i \).

Solving equations demands attention to detail and an understanding of different mathematical objects, including complex numbers, when the situation requires it. Regular practice with various types of equations will enhance understanding and skill in finding solutions.