Problem 47 - Elementary Algebra

Question

For the following problems, solve the equations, if possible. $$ a^{2}=9 $$

Step-by-Step Solution

Answer

Answer: The possible values of \(a\) are 3 and -3.

Step 1: Identify the square root of both sides of the equation

Since \(a^2 = 9\), we can take the square root of both sides to begin isolating the variable. Remember that when taking the square root of a square, you also need to consider both the positive and negative possibilities of the root.

Step 2: Calculate the square root

Calculate the square root by applying the square root function on both sides of the equation. Keep in mind that the square root of any number squared will be the number itself, so we get: $$ \sqrt{a^2} = \pm \sqrt{9} $$

Step 3: Simplify the equation

Simplify the equation by solving the square root of 9, which results in: $$ a = \pm 3 $$ Since a quadratic equation can have two possible solutions, in this case, \(a\) can be either \(3\) (positive root) or \(-3\) (negative root).

Key Concepts

Understanding Square Roots

The square root is a mathematical concept used to find the original number that was squared. In simpler terms, if you know a number's square, a square root helps you find the number itself. For example, in this problem, we dealt with the equation \( a^2 = 9 \). Here, 9 is known as a 'perfect square,' meaning it results from squaring an integer. To solve for \( a \), we take the square root of both sides.

• The square root of 9 is 3, because \( 3 \times 3 = 9 \).
• In mathematical notation, this is represented as \( \sqrt{9} \).

Using the square root is crucial in simplifying equations where variables are squared.

Exploring Positive and Negative Roots

When you take the square root of a squared number, like \( a^2 = 9 \), there are always two potential solutions: one positive and one negative. This concept is important because squaring any number, whether positive or negative, gives a positive result. So, both \( 3 \) and \( -3 \) squared equal 9.

If you spot \( \sqrt{a^2} = \pm \sqrt{9} \) in the solution, the plus-minus (\( \pm \)) symbol indicates that both \( a = 3 \) and \( a = -3 \) are correct answers.

Always remember:

• Squaring eliminates the sign of the number.
• Positive and negative roots ensure you consider all possible values for the variable.

This understanding prevents missing out on any potential solutions.

Solving Equations Step-by-Step

When solving an equation like \( a^2 = 9 \), it's essential to follow a step-by-step approach to avoid errors and gain a clear understanding of what you're doing. Here's how this unfolds:

• Step 1: Identify the square root operation. Realize you need to find which number, when squared, equals 9.
• Step 2: Apply the square root to both sides. You ensure that both sides are treated equally, leading to \( \sqrt{a^2} = \pm \sqrt{9} \).
• Step 3: Simplify the result. Calculate \( \sqrt{9} \) to find 3. This gives you both potential solutions: \( a = 3 \) and \( a = -3 \).

By following these steps, you systematically break down the problem and solve it, ensuring no steps are skipped.