Problem 65 - Algebra 2

Question

Solve each equation by completing the square. $$ x^{2}-6 x+18=0 $$

Step-by-Step Solution

Answer

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

Step 1: Move Constant to Right Side

Start by moving the constant term to the right side of the equation.\[ x^2 - 6x + 18 = 0 \]Subtract 18 from both sides to get: \[ x^2 - 6x = -18 \]

Step 2: Determine the Value to Complete the Square

To complete the square, take the coefficient of the linear term (\(-6\)), divide it by 2, and then square it.\[ \left( \frac{-6}{2} \right)^2 = 9 \]Add and subtract this value on the left side of the equation to maintain equality. This becomes: \[ x^2 - 6x + 9 - 9 = -18 \]Rewrite it as: \[ (x^2 - 6x + 9) - 9 = -18 \]

Step 3: Rewrite as a Perfect Square

The expression inside the parenthesis is now a perfect square trinomial. Rewrite it as:\[ (x - 3)^2 = -18 + 9 \]Simplifying the right side gives:\[ (x - 3)^2 = -9 \]

Step 4: Solve for x by Taking the Square Root

Take the square root of both sides of the equation. Remember to consider both positive and negative roots due to the square.\[ \sqrt{(x-3)^2} = \pm \sqrt{-9} \]This simplifies to: \[ x - 3 = \pm 3i \]

Step 5: Isolate the Variable

Solve for \(x\) by adding 3 to both sides of the equation.\[ x = 3 \pm 3i \]The solutions are \(x = 3 + 3i\) and \(x = 3 - 3i\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two and can be written in the standard form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The variable \(x\) represents the unknown we are solving for. Quadratic equations can be solved using different methods, such as factoring, the quadratic formula, and completing the square.

The method of completing the square involves rearranging the equation into a perfect square trinomial, making it easier to solve for \(x\). This approach is especially helpful when the quadratic cannot be easily factored.

Completing the square helps reveal the equation's structure, providing insights into the nature of its roots, such as real or complex numbers.

Imaginary Numbers

Imaginary numbers form part of the complex number system and come into play when dealing with the square root of negative numbers. The imaginary unit is denoted by \(i\), where \(i = \sqrt{-1}\).

When we solve quadratic equations by completing the square, we may find that taking the square root of a negative number is necessary, as seen in the problem where \((x-3)^2 = -9\).

To handle this, we express \(\sqrt{-9}\) as \(3i\), whereby the solution involves incorporating \(i\). Thus, the roots of our equation include imaginary numbers: \(3 + 3i\) and \(3 - 3i\).

Imaginary numbers are essential for solving equations that do not have real solutions and are crucial in many engineering and physics applications.

Perfect Square Trinomial

A perfect square trinomial results from squaring a binomial, and it takes the form \((x + a)^2 = x^2 + 2ax + a^2\) or \((x - a)^2 = x^2 - 2ax + a^2\). In the equation we are solving, by completing the square, we form the expression \((x - 3)^2\).

This expression arises after manipulating \(x^2 - 6x + 9\), which clearly fits the pattern of a perfect square trinomial. The step of completing the square involves finding this structure, which simplifies the quadratic into a form that can be easily solved using the square root method.

Understanding how to recognize and generate perfect square trinomials is a key skill in algebra, as it allows for efficient problem solving and can make complex quadratic equations more manageable.

Solving Algebraic Equations

Solving algebraic equations involves finding the values of the unknown variable(s) that make the equation true. For quadratic equations, there are multiple strategies available.

Completing the square is a method that restructures the equation into a perfect square trinomial, making it straightforward to solve by extracting the square roots. This requires careful calculation, such as taking half of the linear coefficient, squaring it, and adjusting the equation accordingly.

Once in the form \((x - a)^2 = b\), solving involves:

• Taking the square root of both sides.
• Remembering to consider both positive and negative roots (\(\pm\)).
• Isolating the variable \(x\) to find the solutions.

Such systematic approaches ensure that all potential solutions are found, whether they are real or involve imaginary numbers, as seen in the original exercise.