Problem 28 - Algebra A Combined Function

Question

Solve. $$ x^{2}=9 $$

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -3\).

Step 1: Identify the Equation Type

The given equation is a quadratic equation since it is in the form of \(x^2 = 9\) and has one variable squared.

Step 2: Square Root Both Sides

To solve the equation \(x^2 = 9\), we need to take the square root of both sides. This gives us two potential solutions: \(x = \sqrt{9}\) and \(x = -\sqrt{9}\).

Step 3: Simplify the Square Roots

Calculate the square root of 9, which is 3. This gives us the two potential solutions: \(x = 3\) and \(x = -3\).

Step 4: Verify the Solutions

To ensure correctness, substitute each solution back into the original equation. For \(x = 3\), \((3)^2 = 9\), and for \(x = -3\), \((-3)^2 = 9\). Both solutions satisfy the original equation.

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we want to solve for. These types of equations can appear in different forms, but the essential aspect is the \(x^2\) term.In this particular exercise, we focus on a simpler form of a quadratic equation, \(x^2 = 9\). The absence of \(bx\) and \(c\) terms makes this equation easier to solve. The challenge mainly lies in applying the correct method to find the values of \(x\) that make the equation true. In general, solving quadratic equations requires identifying the equation type and then applying a suitable method, such as factoring, completing the square, using the quadratic formula, or, as in this case, the square root method.

Square Root Method

The square root method is an efficient approach for solving quadratic equations of the form \(x^2 = k\), where \(k\) is a constant. This method is straightforward because it involves fewer steps. To solve using this method:

• Take the square root of both sides of the equation. This acknowledges that squaring a number results in its absolute value, leading to two possible solutions: a positive and a negative one.
• For the equation \(x^2 = 9\), taking the square root of both sides gives us \(x = \sqrt{9}\) and \(x = -\sqrt{9}\).
• Simplify the results: \(\sqrt{9} = 3\), thus the solutions are \(x = 3\) and \(x = -3\).

This method highlights why a quadratic equation typically has two solutions. Remember, this method works best for equations where the variable squared is isolated on one side.

Verification of Solutions

Verification is a critical part of solving equations. It involves substituting each potential solution back into the original equation to ensure it satisfies the equation.For our problem, substitute \(x = 3\) and \(x = -3\) back into \(x^2 = 9\):

• For \(x = 3\): Substitute into the equation to get \((3)^2 = 9\). This is true, meaning \(x = 3\) is a valid solution.
• For \(x = -3\): Substitute into the equation to get \((-3)^2 = 9\). This too is correct, confirming \(x = -3\) as a solution.

This process confirms both solutions, ensuring that no calculation errors were made, and the solutions correctly solve the original equation. Verification reinforces understanding and builds confidence in the solution correctness.