Problem 17 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by completing the square. $$n^{2}=6 n-4$$

Step-by-Step Solution

Answer

The solutions are \(n = 3 + \sqrt{5}\) and \(n = 3 - \sqrt{5}\).

Step 1: Move the Constant

First, we need to rearrange the equation to make it easier to complete the square. Start with the given equation: \[n^2 = 6n - 4\] Move the constant term to the other side of the equation: \[n^2 - 6n = -4\]

Step 2: Prepare to Complete the Square

Identify the coefficient of the linear term (the term with \(n\)). This coefficient is \(-6\). We will use this to complete the square in the next step.

Step 3: Calculate the Value to Complete the Square

Take half of the coefficient from the linear term and square it. The linear term's coefficient is \(-6\), so:\[\left(\frac{-6}{2}\right)^2 = 9\]Add and subtract this value on the left side to complete the square:\[n^2 - 6n + 9 = -4 + 9\]

Step 4: Rewrite as a Perfect Square

The left side can now be rewritten as a perfect square trinomial:\[(n - 3)^2 = 5\]This is because \((n - 3)^2\) expands to \(n^2 - 6n + 9\).

Step 5: Solve the Equation

To solve for \(n\), take the square root of both sides:\[n - 3 = \pm \sqrt{5}\]This gives two equations:\[n - 3 = \sqrt{5}\] and\[n - 3 = -\sqrt{5}\]

Step 6: Solve for \(n\)

Solve each of the two equations for \(n\):1. \(n = 3 + \sqrt{5}\)2. \(n = 3 - \sqrt{5}\)These are the final solutions to the equation.

Key Concepts

Completing the Square

Completing the square is a useful method in algebra to solve quadratic equations. It transforms a quadratic equation into a perfect square trinomial. This makes it easier to solve for the variable. Let's explore this process step by step using our example equation: \[n^2 = 6n - 4\].

Initially, we rearrange the equation so that all terms involving the variable \(n\) are on one side, and the constant is on the other: \[n^2 - 6n = -4\]. This setup allows us to complete the square.

• Identify the linear coefficient, which is the number in front of \(n\), in this case, \(-6\).
• Take half of this coefficient and square it: \(\left( \frac{-6}{2} \right)^2 = 9\). Add and subtract this square to form a perfect square trinomial.
• The expression becomes \(n^2 - 6n + 9 = 5\), which can be rewritten as \( (n - 3)^2 = 5 \).

By transforming the equation into this format, it becomes super straightforward to solve.

Solving Equations

After you've rewritten a quadratic equation in the form of a perfect square like \((n - 3)^2 = 5\), solving the equation becomes much simpler. The next step is to solve for the variable by taking the square root of both sides.

Keep in mind:

• When you take the square root of both sides, you should include both the positive and negative roots. In our example, \((n - 3)^2 = 5\), the solutions become \(n - 3 = \pm \sqrt{5}\).
• Separate this into two different linear equations: \(n - 3 = \sqrt{5}\) and \(n - 3 = -\sqrt{5}\).
• Solve each equation individually by adding 3 to both sides, resulting in \(n = 3 + \sqrt{5}\) and \(n = 3 - \sqrt{5}\).

These solutions are the values of \(n\) that satisfy the original equation.

As you can see, by writing the quadratic as a perfect square, the process of solving becomes straightforward.

Algebraic Techniques

Algebraic techniques like completing the square are crucial for solving quadratic equations. They're part of a toolkit every math student should develop. Here are some key algebraic strategies involved in this process:

• Rearranging Equations: This involves moving terms around so that variables are isolated. In our example, this meant shifting the equation to \[n^2 - 6n = -4\].
• Identifying Coefficients: Recognizing the linear term's coefficient helps in determining what number to add and subtract for completing the square. This coefficient is key to forming perfect square trinomials.
• Balancing Equations: When you add a number to one side of the equation, you must add it to the other as well to maintain balance. With \[n^2 - 6n = -4\], adding and subtracting 9 ensures the equation remains equal.

Mastering these techniques will make solving quadratic equations much smoother and allows problems to be tackled efficiently in a structured manner.