Problem 25 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$16-x^{2}=0$$

Step-by-Step Solution

Answer

Factor the expression to find the solution: \[x^2 - 16 = (x-4)(x+4) = 0\] Thus, the solutions are \(x = 4\) or \(x = -4\).

Step 1: Factor using difference of squares

\(16 - x^2 = (4-x)(4+x) = 0\)

Step 2: Solve

\(4-x = 0\) gives \(x = 4\), and \(4+x = 0\) gives \(x = -4\).
The solutions are \(x = 4\) and \(x = -4\).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and come in the standard form of \(ax^2 + bx + c = 0\). They represent a type of polynomial equation of degree 2. When working with quadratic equations, you can solve them using various techniques such as factoring, completing the square, and the quadratic formula.
Understanding the structure is key. For our example, \(16 - x^2 = 0\), we notice it doesn’t initially look like the standard form. However, it still represents a quadratic equation because it has a variable raised to the power of 2. Recognizing equivalent forms, like rearranging it to \(x^2 = 16\), makes solving them more straightforward.
This equation even hints at completing the square, showing how versatile quadratic equations are. Knowing how to navigate between these forms is crucial when picking the right technique to solve them.

Completing the Square

Completing the square is a method for solving quadratic equations by turning part of the equation into a perfect square trinomial. This technique is especially helpful when the quadratic equation cannot be easily factored.
Here's how it works:

• Take the equation \(ax^2 + bx = c\).
• Move the constant term \(c\) to the other side.
• Add the square of half the coefficient of \(x\) to both sides to create a perfect square on one side of the equation.
• For example, if you have something like \(x^2 - 6x\), you add \((3)^2 = 9\), resulting in \((x-3)^2\).

In our specific problem \(x^2 = 16\), this method isn’t necessary because it can be easily solved by taking the square root. Yet learning it prepares you for more challenging problems. Completing the square also connects seamlessly to deriving the quadratic formula.

Solving Equations

Solving algebraic equations involves finding the values of the variables that make the equation true. While linear equations are quite straightforward, quadratic equations like \(x^2 = 16\) need a bit more insight.
To solve this equation:

• First, recognize it as a difference of squares, \(16 = (4)^2\), and factor it if needed.
• You can also solve it by isolating \(x\) and taking the square root of both sides.
• So, you think of the equation as \(x^2 = (4)^2\) which leads to \(x = \pm 4\).

Understanding these steps is crucial in ensuring you find all possible solutions. Quadratic equations can often have two solutions because the square of both positive and negative numbers results in a positive number.