Problem 89 - Algebra 1: Concepts and Skills

Question

Evaluate \(3 \pm \sqrt{(-3)^{2}-4(0.5)(-8)}\)

Step-by-Step Solution

Answer

The answer to the exercise is two values, 8 and -2.

Step 1: Solving the equation within the sqrt

First, we need to know the value inside the sqrt which is \((-3)^{2}-4(0.5)(-8)\). Simplify this to get 9 - (-16), which simplifies to 9 + 16 = 25.

Step 2: Evaluating Square root

Square root of 25 is ±5. The reason that we take both the positive and negative sqrt is due to the ± symbol in the given exercise.

Step 3: Adding and subtracting the sqrt result from 3

Now that we know sqrt of 25 is ±5, we substitute this back into the exercise. So this gives \(3 \pm 5\). This may be seen as two equations, 3 + 5 = 8 and 3 - 5 = -2.

Key Concepts

Evaluating Square Roots

When dealing with square roots, it’s essential to understand what they represent. A square root asks the question: "What number multiplied by itself gives us this number?" For instance, the square root of 25 is 5 because 5 squared (5 x 5) equals 25.

But some exercises, like the one we're discussing, might require considering both the positive and negative square roots. This is because both 5 and -5, when squared, result in 25. Hence, we get two possibilities: \(\pm 5\).

• Positive root: This is usually the more straightforward solution, giving us the expected result.
• Negative root: Important in cases where solutions can have multiple forms, especially in quadratic equations.

Embracing both roots ensures that all solutions are considered, which is crucial in solving real-world math problems.

Simplifying Expressions

Simplifying expressions is a crucial step in solving math problems. By breaking down expressions into simpler parts, we can solve them more easily.

In our example, we begin with the expression inside the square root: \((-3)^2 - 4(0.5)(-8)\). Step by step, this means:

• \((-3)^2 = 9\): Squaring -3 gives us 9.
• \(4(0.5)(-8) = -16\): This operation involves multiplication, taking care of the negative sign.
• Finally, 9 + 16 = 25. This completes the expression simplification inside the square root.

Finding this simplified form turns complex problems into easier steps. It ensures accuracy and clarity in reaching the correct solution.

Positive and Negative Solutions

In many algebra problems, especially those involving square roots or quadratic equations, we encounter "+" and "-" signs indicating multiple solutions.

This part involves understanding that a given value can lead to multiple outcomes. When we plug in \(+5\) or \(-5\) from our square root, we develop:

• \(3 + 5 = 8\): This is one solution where the positive root is added.
• \(3 - 5 = -2\): This uses the negative root, resulting in a different outcome altogether.

Both results are valid, showcasing how variables react to different operations. Remember: embracing both positive and negative solutions provides the complete set of answers needed for a comprehensive understanding.