Problem 61 - Algebra 1: Concepts and Skills

Question

Choose a method and solve the quadratic equation. Explain your choice. $$ x^{2}-9=0 $$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-9=0\) are \(x=3\) and \(x=-3\).

Step 1: Identify the Quadratic Equation

The quadratic equation is \(x^{2}-9=0\) .

Step 2: Factoring the Quadratic Equation

The equation is a difference of squares. Every difference of squares can be factored as \(a^{2}-b^{2}=(a-b)(a+b)\). So, for our equation, \(x^{2} = (x)^{2}\) and 9 = (3)^{2}. So factoring gives us \((x-3)(x+3) = 0\).

Step 3: Solve for x

To solve for x, set each factor equal to zero and solve. This gives us \(x-3=0\) and \(x+3=0\). Solving for x, we get \(x=3\) and \(x=-3\).

Key Concepts

Factoring Quadratic Equations

Understanding how to factor quadratic equations is crucial for solving them effectively. A quadratic equation is typically in the form of ax^2 + bx + c = 0.

Factoring involves breaking down the quadratic equation into a product of simpler expressions that, when multiplied together, yield the original quadratic expression. This process often requires finding two numbers that not only multiply to give the constant term, c, but also add up to b, the coefficient of the x term. In the exercise given, we recognize that the equation x^2 - 9 is missing the bx term, suggesting it might be a special form where factoring can be straightforward.

Using this method allows us to convert a complex problem into a simpler one that we can solve by using additional properties such as the zero product property which will be discussed later in this article. Factoring is a preferred method when the quadratic equation can be easily decomposed into simpler binomial factors.

Difference of Squares

The difference of squares is a specific type of factoring that applies when you have a subtraction between two squared terms. It is recognizable by the pattern a^2 - b^2. According to this formula, you can factor such an expression into (a + b)(a - b).

Take the exercise x^2 - 9 = 0 as an example; we can identify x^2 as a^2 and 9 as b^2, where in this case, a = x and b = 3 (since 9 is 3^2). The equation simplifies to (x + 3)(x - 3) = 0. This method is quick and effective when the quadratic falls into the difference of squares pattern, as it allows us to bypass completing the square or using the quadratic formula, which can be more complex.

Zero Product Property

Once a quadratic equation is factored, the zero product property can be used to find the solutions. It states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if ab = 0, then a = 0 or b = 0, or both.

Looking again at the equation from the exercise, we have (x - 3)(x + 3) = 0. Applying the zero product property, we set each factor equal to zero: x - 3 = 0 and x + 3 = 0. Solving these linear equations separately gives us the solutions x = 3 and x = -3. The zero product property is a powerful tool in solving quadratic equations because it reduces the problem to solving simpler linear equations.