Problem 16 - Algebra 1

Question

Choose a method to solve the quadratic equation. What method did you use? Explain your choice. $$x^{2}-9=0$$

Step-by-Step Solution

Answer

The roots of the equation \(x^{2}-9=0\) are \(x = -3\) and \(x = 3\). The method used to solve this is identifying and applying the difference of squares.

Step 1: Identify Type of Quadratic Equation

The given quadratic equation is \(x^{2}-9=0\). This is a special type of quadratic equation known as a difference of two squares, as 9 is a perfect square. The standard form of a difference of two squares is \(a^{2} - b^{2} = (a+b)(a-b)\). This allows us to use a simplification for finding the roots.

Step 2: Apply the Difference of Squares

Using the formula, we rewrite the equation \(x^{2}-9=0\) as \((x+3)(x-3)=0\)

Step 3: Solve for x

Now we set each binomial expression equal to zero and solve for x, which gives us \(x+3 = 0 \Rightarrow x = -3\) and \(x-3 = 0 \Rightarrow x = 3\)

Key Concepts

Difference of Two Squares

Understanding the concept of the difference of two squares is essential when solving certain quadratic equations. This mathematical trick can turn a complex-looking problem into a simpler one that's easier to solve. It applies to any quadratic equation of the form \(a^2 - b^2 = 0\), where \(a\) and \(b\) are both perfect squares. The equation then can be broken down into two binomials, \(a+b\) and \(a-b\), that are multiplied: \(a^2 - b^2 = (a+b)(a-b)\).

For example, in the exercise \(x^2 - 9 = 0\), recognizing that \(9\) is a perfect square (as it is \(3^2\)) allows us to identify the equation as a difference of two squares. Thus, we could express it as \(x^2 - 3^2 = 0\), which readily factors into \(x + 3)(x - 3) = 0\). This method simplifies the process of finding roots, as it breaks the problem down into two easier equations.

Factoring Quadratics

Factoring quadratics is a widely-used technique for solving quadratic equations that can be written in a standard form of \(ax^2 + bx + c = 0\). The goal is to break down the quadratic into a product of two binomials. When the coefficients \(a\), \(b\), and \(c\) are such that the quadratic is factorable over the integers, we look for two numbers that multiply to \(ac\) and add up to \(b\).

However, the given exercise \(x^2 - 9 = 0\) presents a simpler scenario where \(a = 1\), \(b = 0\), and \(c = -9\). In this case, because the middle term \(bx\) is absent, we can immediately identify the equation as a difference of two squares and factor it without further steps. Factoring is a powerful method, especially useful when the quadratic is easily decomposable into integer factors.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. It states that the solutions for \(ax^2 + bx + c = 0\) are given by \(x = {-b \pm \sqrt{b^2-4ac}} / {2a}\). It serves as a universal solution technique when factoring is not straightforward or possible.

The formula is derived from completing the square and is applicable to all quadratics, providing an exact answer even when the roots are not rational numbers. While not necessary for the given exercise \(x^2 - 9 = 0\), since we can use the difference of two squares, the quadratic formula is a crucial fallback solution when dealing with more complex equations.

Roots of Quadratic

The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. They are also known as the solutions or zeroes of the equation. Depending on the quadratic's coefficients, there can be two distinct real roots, one real repeated root, or two complex roots.

In the context of the exercise \(x^2 - 9 = 0\), after factoring the equation using the difference of two squares, we ended up with two binomial expressions. Setting each binomial equal to zero \( (x + 3)(x - 3) = 0\) and solving for \(x\) gives us the roots \(x = -3\) and \(x = 3\). This illustrates an instance where we have two distinct real roots, which are particularly straightforward to identify and validate. Understanding how to find the roots is fundamental in grasping the nature of quadratic equations.