Distance Between Points & Quadratic Equation Roots: Solved!

Hey guys! Let's dive into some cool math problems today, focusing on finding the distance between two points and figuring out the value of 'k' in a quadratic equation. We'll break it down step by step so it's super easy to follow. Buckle up, math enthusiasts!

Finding the Distance Between Two Points

Let's tackle the first problem: finding the distance between the points (2, 3) and (4, 1). This is a classic coordinate geometry problem, and we'll use the distance formula to solve it. Remember, the distance formula helps us calculate the length of a line segment connecting two points in a coordinate plane. It's derived from the Pythagorean theorem, which you might remember from geometry class. The distance formula is a fundamental concept in coordinate geometry and has wide applications in various fields, including physics, engineering, and computer graphics. Understanding how to use it can greatly enhance your problem-solving skills in mathematics and related disciplines. Applying this formula correctly is crucial for accurate calculations in various mathematical and real-world problems.

Understanding the Distance Formula

The distance formula is given by:

√[(x₂ - x₁)² + (y₂ - y₁)²]

Where:

• (x₁, y₁) are the coordinates of the first point
• (x₂, y₂) are the coordinates of the second point

This formula might look a bit intimidating at first, but it's really straightforward once you break it down. The formula essentially calculates the length of the hypotenuse of a right triangle formed by the two points and their projections onto the x and y axes. This is where the connection to the Pythagorean theorem becomes clear, as the distance formula is essentially an application of the theorem in a coordinate plane. Understanding the derivation of the formula can help you remember it and apply it confidently in different situations. Additionally, recognizing the formula's connection to the Pythagorean theorem reinforces your understanding of fundamental geometric principles.

Applying the Formula to Our Points

In our case, we have:

• (x₁, y₁) = (2, 3)
• (x₂, y₂) = (4, 1)

Now, let's plug these values into the distance formula: The key to successfully applying the distance formula lies in correctly identifying the coordinates of the two points and substituting them into the formula. Careful attention to the order of operations (subtraction, squaring, and square root) is also important to avoid errors. Practice with various examples can help you master the technique and build confidence in your calculations. Remember to double-check your work, especially when dealing with negative numbers or complex expressions.

√[(4 - 2)² + (1 - 3)²]

Step-by-Step Calculation

1. Calculate the differences:
   • 4 - 2 = 2
   • 1 - 3 = -2
2. Square the differences:
   • 2² = 4
   • (-2)² = 4
3. Add the squares:
   • 4 + 4 = 8
4. Take the square root:
   • √8 = √(4 * 2) = 2√2

So, the distance between the points (2, 3) and (4, 1) is 2√2 units. And that's our answer! The step-by-step calculation allows us to break down the problem into manageable chunks, making it easier to understand and solve. This approach is particularly helpful when dealing with more complex problems or when you're just starting to learn the concept. By systematically working through each step, you can minimize the chances of making errors and gain a deeper understanding of the underlying principles.

Finding the Value of 'k' in a Quadratic Equation

Next up, let's tackle the second problem: finding the value of 'k' in the quadratic equation 3x² - 12x + k = 0, given that the roots are equal. This problem involves understanding the discriminant of a quadratic equation. The discriminant plays a crucial role in determining the nature of the roots of a quadratic equation. It tells us whether the roots are real and distinct, real and equal, or complex. Understanding the discriminant is essential for solving quadratic equations and analyzing their solutions. Furthermore, the concept of the discriminant extends to higher-degree polynomials and has applications in various areas of mathematics and engineering.

Understanding the Discriminant

For a quadratic equation in the standard form ax² + bx + c = 0, the discriminant (D) is given by:

D = b² - 4ac

The discriminant tells us about the nature of the roots:

• If D > 0, the equation has two distinct real roots.
• If D = 0, the equation has two equal real roots (or one repeated real root).
• If D < 0, the equation has no real roots (two complex roots).

In our case, we are given that the roots are equal, which means the discriminant must be zero. The discriminant is a powerful tool for quickly assessing the nature of the roots without actually solving the equation. This can save time and effort in many problem-solving scenarios. Additionally, the discriminant provides valuable insights into the graphical representation of the quadratic equation, as it relates to the number of x-intercepts of the parabola. Understanding these connections enhances your overall understanding of quadratic equations and their properties.

Applying the Discriminant to Our Equation

Our equation is 3x² - 12x + k = 0. Comparing this with the standard form ax² + bx + c = 0, we have:

• a = 3
• b = -12
• c = k

Since the roots are equal, D = 0. Let's plug the values into the discriminant formula: Identifying the coefficients a, b, and c correctly is the first crucial step in applying the discriminant. Pay close attention to the signs and values to avoid errors. Once you have identified the coefficients, substituting them into the formula and simplifying the expression will lead you to the solution. Remember that the condition D = 0 is specific to the case where the roots are equal, and different conditions apply for distinct or complex roots.

0 = (-12)² - 4 * 3 * k

Solving for 'k'

1. Simplify the equation:
   • 0 = 144 - 12k
2. Rearrange the equation to isolate 'k':
   • 12k = 144
3. Divide both sides by 12:
   • k = 144 / 12
   • k = 12

Therefore, the value of k is 12. Nicely done! Solving for 'k' involves algebraic manipulation, which is a fundamental skill in mathematics. Understanding the properties of equality and applying them correctly is essential for solving equations of all types. Practice with various examples can help you develop your algebraic skills and build confidence in your problem-solving abilities. Remember to always check your solution by substituting the value of 'k' back into the original equation to ensure it satisfies the given conditions.

Conclusion

So, guys, we've successfully solved two pretty cool math problems today! We found the distance between two points using the distance formula and figured out the value of 'k' in a quadratic equation by using the discriminant. These are important concepts in coordinate geometry and algebra, and mastering them will definitely help you in your math journey. Remember, practice makes perfect, so keep solving problems and exploring the wonderful world of mathematics! Keep practicing and you'll become a math whiz in no time! You've got this! These types of problems are not just limited to textbooks; they often appear in standardized tests and have real-world applications in fields like engineering, physics, and computer science. Mastering these concepts will not only help you academically but also prepare you for future challenges in various career paths.