Problem 16 - 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 are \(x = 3\) and \(x = -3\).

Step 1: Identify the Equation Type

Recognize that the equation \(x^{2} - 9 = 0\) can be seen as a perfect square. That is to say, we could express it as \(x^{2} - 3^{2} = 0\).

Step 2: Apply the Difference of Squares Formula

To solve this type of equations, we can apply the difference of squares formula: \(a^{2} - b^{2} = (a - b)(a + b)\). Applying this formula here gives: \(x^{2} - 3^{2} = (x - 3)(x + 3)\). Therefore, the equation can be rewritten as \( (x - 3)(x + 3) = 0\).

Step 3: Solve for x

We now have two simple equations to solve: x - 3 = 0 and x + 3 = 0. Solving these for x gives us x = 3 and x = -3.