Problem 17 - College Algebra with Corequisite Support

Question

For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers. (-4,1) and (3,-4)

Step-by-Step Solution

Answer

The distance is \(\sqrt{74}\).

Step 1: Identify Coordinates

First, identify the coordinates of the two given points. Point 1 is \((-4, 1)\) and Point 2 is \( (3, -4) \).

Step 2: Apply the Distance Formula

To find the distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Step 3: Substitute Coordinates Into Formula

Substitute the coordinates of the points into the formula: \(x_1 = -4, \, y_1 = 1, \, x_2 = 3, \, y_2 = -4\)\[d = \sqrt{(3 - (-4))^2 + (-4 - 1)^2}\]

Step 4: Simplify Inside the Square Root

First, calculate the differences: \(3 - (-4) = 3 + 4 = 7\) and \(-4 - 1 = -5\). Plug these into the formula: \(d = \sqrt{7^2 + (-5)^2} = \sqrt{49 + 25}\)

Step 5: Calculate the Sum and Simplify the Square Root

Add the squares: \(49 + 25 = 74\). Then \(d = \sqrt{74}\).

Step 6: Express in Simplest Radical Form

Simplify \(\sqrt{74}\). Since 74 is 37 times 2 and neither is a perfect square, \(\sqrt{74}\) is already in its simplest radical form.

Key Concepts

Simplifying Radicals

Simplifying radicals is a key skill when working with mathematical problems, especially when dealing with distances in the coordinate plane. When we talk about a radical, we are referring to expressions under a square root sign, like \( \sqrt{74} \). Our main goal is to break this down into simpler parts if possible. This process helps make mathematical expressions neater and often more understandable.

To simplify a radical, you need to factor the number inside the square root into its prime factors. For \( \sqrt{74} \), this means identifying 74 as the product of 37 and 2, both of which are prime numbers. Since neither of these numbers is a perfect square, there is no simpler radical form, indicating that \( \sqrt{74} \) is already as simplified as possible.

Here are a few tips for simplifying radicals:

• Factor numbers into primes to see if any can be grouped into perfect squares.
  For instance, in \( \sqrt{18} \), which factors as 2 and 3 squared, simplify to \( 3\sqrt{2} \) because \( \sqrt{9} = 3 \).
• If the number inside the radical isn't a perfect square or no factors are perfect squares, the radical is already simplified.
  Think of radicals in their most compact state for clarity and precision.

Coordinate Plane

The coordinate plane is a crucial component in understanding how to calculate distances and visualize problems in algebra. Named after René Descartes, the coordinate plane (or Cartesian plane) uses two perpendicular lines, the x-axis, and the y-axis, to define a 2D space.

In the coordinate plane, each point is represented as \( (x, y) \), where 'x' is the horizontal position and 'y' is the vertical position. By using this system, you can easily determine the distance between any two points using the distance formula, which is derived from the Pythagorean theorem. The points \( (-4, 1) \) and \( (3, -4) \) in the original problem are just such examples.

The coordinate plane aids in:

• Visualizing problems, making it easier to comprehend geometric relationships.
• Calculating distances or areas more effectively without complex diagrams.
• Working out algebraic problems that involve spatial reasoning or need a clear representation.

Algebra Problems

Algebra problems often involve finding unknown values or expressions that satisfy certain conditions. For example, calculating the distance between two points based on their coordinates is a common type of algebra problem, which incorporates the use of formulas and simplification techniques.

These problems typically follow a structured approach:

• Identify the necessary variables or elements from the problem, as seen where coordinates were labeled as \( (x_1, y_1) \) and \( (x_2, y_2) \).
• Apply known formulas or methods to derive an equation, referenced as the distance formula in this case.
• Substitute values and simplify the resulting expression to arrive at the solution. For example, progressing through calculations to simplify \( \sqrt{74} \).

Each of these steps leverages mathematical rules to foster logical thinking and problem-solving skills, essential not only in algebra but in broader applications of mathematics. Understanding how to break down and solve these kinds of problems sets the foundation for tackling more complex mathematical concepts.