Problem 21 - Intermediate Algebra

Question

Use the method of completing the square to solve each quadratic equation. $$ n^{2}-8 n+17=0 $$

Step-by-Step Solution

Answer

The solutions are \( n = 4 + i \) and \( n = 4 - i \).

Step 1: Start with the Original Equation

We begin with the quadratic equation given in the problem: \[ n^2 - 8n + 17 = 0 \] Our goal is to rewrite this equation by completing the square.

Step 2: Move Constant to the Other Side

To complete the square, first move the constant term to the other side of the equation. Subtract 17 from both sides: \[ n^2 - 8n = -17 \]

Step 3: Find the Term to Complete the Square

To complete the square, take the coefficient of the linear term (-8), divide by 2, and then square it. The calculation is: \[ \left(\frac{-8}{2}\right)^2 = 4^2 = 16 \]Add this square to both sides of the equation to maintain equality.

Step 4: Add and Simplify

Add 16 on both sides of the equation:\[ n^2 - 8n + 16 = -17 + 16 \]This simplifies to:\[ n^2 - 8n + 16 = -1 \]

Step 5: Write the Perfect Square Trinomial

The left-hand side of the equation is a perfect square trinomial:\[ (n - 4)^2 = -1 \]Thus, the equation becomes:\[ (n - 4)^2 = -1 \]

Step 6: Solve for n by Taking the Square Root

Take the square root of both sides to solve for \( n \):\[ \sqrt{(n-4)^2} = \sqrt{-1} \]This gives us:\[ n - 4 = \pm i \] (where \( i \) is the imaginary unit).

Step 7: Solve for n

Finally, solve for \( n \) by adding 4 to both sides:\[ n = 4 \pm i \]This gives the two complex solutions for \( n \):\[ n = 4 + i \quad \text{and} \quad n = 4 - i \].

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental pillar of algebra and appear frequently in various fields, such as physics, engineering, and finance. In its standard form, a quadratic equation looks like this:

• ax² + bx + c = 0,

where a, b, and c are constants, and x is the variable. One common method of solving such equations is by completing the square. This technique transforms a quadratic equation into a perfect square trinomial, making it easier to solve.
Completing the square involves several key steps:

• Move the constant term to the opposite side of the equation.
• Find the necessary term to complete the square by taking half of the linear coefficient, squaring it, and adding both sides.
• Rewrite the equation as a perfect square trinomial, allowing you to take square roots easily.

Through this method, we can derive solutions to quadratic equations, whether they are real or complex.

Exploring Complex Solutions

When solving quadratic equations, you may encounter a scenario where the solutions are not real but complex. This situation arises especially when the equation is set equal to a negative number after completing the square. Complex solutions consist of both a real part and an imaginary part. They are generally expressed as:

• x = a ± bi,

where a and b are real numbers, and i denotes the imaginary unit.
In the context of our quadratic equation, completing the square gives us

• (n - 4)² = -1,

indicating non-real solutions. Introducing complex numbers can extend the real number system, capturing the full set of possible solutions to quadratic equations, demonstrating their elegant completenes.

Introduction to Imaginary Numbers

Imaginary numbers may sound abstract, but they form a crucial component of mathematics, particularly when dealing with complex solutions in quadratic equations. The imaginary unit, denoted as i, satisfies the equation:

• i² = -1

This property allows us to handle the square root of negative numbers. In real-number arithmetic, the square root of a negative number is undefined; however, imaginary numbers provide a complete solution.
For example, in the equation

• (n - 4)² = -1

solving for n yields:

• n - 4 = ±i,

where ±i represents the imaginary parts of the solution. Understanding imaginary numbers unlocks the full potential of solving quadratic equations, providing insight into many real-world applications.