Problem 15 - Basic Technical Mathematics with Calculus

Question

In Exercises \(11-30,\) solve the given quadratic equations by completing the square. Exercises \(11-14\) and \(17-20\) may be checked by factoring. $$n^{2}=4 n-2$$

Step-by-Step Solution

Answer

The solutions for \(n\) are \(2 + \sqrt{2}\) and \(2 - \sqrt{2}\).

Step 1: Move constant to the other side

Rewrite the equation so that all terms involving \(n\) are on one side and the constant is on the other side. Start with the equation \(n^2 = 4n - 2\). Move the \(-2\) to the other side to get: \(n^2 - 4n = -2\).

Step 2: Prepare to complete the square

To complete the square, focus on the \(n^2 - 4n\) expression. Identify the coefficient of \(n\), which is \(-4\), halve it, and then square it. Halving \(-4\) gives \(-2\), and squaring \(-2\) gives \(4\). Add and subtract this square on the left side to maintain balance: \(n^2 - 4n + 4 - 4 = -2\).

Step 3: Simplify

We added \(4\) to help complete the square. Now, simplify by grouping \(n^2 - 4n + 4\) into a perfect square trinomial. This means rewriting \(n^2 - 4n + 4\) as \((n - 2)^2\). Combine terms on the right: \((n - 2)^2 - 4 = -2\).

Step 4: Isolate the perfect square

To solve for \((n - 2)^2\), move \(-4\) to the right side by adding 4 to both sides: \((n - 2)^2 = -2 + 4\). This simplifies to \((n - 2)^2 = 2\).

Step 5: Solve for n

To solve for \(n\), take the square root of both sides of the equation: \(n - 2 = \pm \sqrt{2}\). Finally, add 2 to both sides to isolate \(n\): \(n = 2 \pm \sqrt{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest power of the variable is two. These equations typically have the general form:

• \(ax^2 + bx + c = 0\)

where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. Quadratic equations are fundamental in algebra and have a characteristic parabola shape when graphed. They can model various real-world situations, from projectile motion to optimization problems.
Understanding quadratic equations involves a few key methods for finding their solutions, such as factoring, using the quadratic formula, and completing the square. Each approach offers different insights and is useful depending on the particular quadratic equation. In our exercise, we used the method of completing the square to solve the quadratic equation. This approach offers a straightforward pathway to solutions by transforming the equation into a perfect square trinomial, simplifying the solving process.

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial that results from squaring a binomial. It takes the form:

• \((a - b)^2 = a^2 - 2ab + b^2\)
• Or \((a + b)^2 = a^2 + 2ab + b^2\)

In our exercise, we converted the expression \(n^2 - 4n + 4\) into \((n - 2)^2\), demonstrating the concept of a perfect square trinomial. To do this, we identified the coefficient of the linear term \(-4\), halved it to find \(-2\), then squared it to add and subtract 4 within the equation to maintain balance.
This process allows us to rewrite the quadratic as a squared term, simplifying the equation and making it easier to solve. Recognizing and forming perfect square trinomials is a powerful algebraic tool, especially in solving quadratic equations by completing the square. It transforms the equation into a more manageable form and directly relates to solving for the variable.

Solving Equations

Solving equations in algebra involves finding the values of the variables that make the equation true. When it comes to quadratic equations, several methods can assist, including completing the square as illustrated in the exercise. After forming a perfect square trinomial, the next step is to take advantage of the properties of squares.
By rewriting the equation in square form, for instance, \((n - 2)^2 = 2\), we simplify the process by taking the square root of both sides. This step introduces a crucial aspect of solving equations: considering both the positive and negative roots. Thus, solving \((n - 2)^2 = 2\) results in two possible solutions, \(n - 2 = \sqrt{2}\) and \(n - 2 = -\sqrt{2}\).
Finally, isolate the variable by performing simple algebraic manipulations—adding or subtracting terms as necessary. Here, adding 2 to both sides produced the solutions \(n = 2 + \sqrt{2}\) and \(n = 2 - \sqrt{2}\). This complete process of solving equations through completing the square not only provides the solutions but deepens one's understanding of algebraic structures.