Proving Real And Distinct Roots Of Quadratic Equation Kx^2 + 5x - 2k = 0

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Hey guys! Today, we're diving into a classic math problem that involves proving the nature of roots for a quadratic equation. Specifically, we're going to show that the roots of the equation kx² + 5x - 2k = 0 are real and distinct for all positive values of k, where k ≠ 0. This is a fundamental concept in algebra, and understanding it will help you tackle similar problems with confidence. So, let's break it down step by step!

Understanding the Discriminant

First off, let's talk about the discriminant. The discriminant is a crucial part of the quadratic formula that tells us about the nature of the roots of a quadratic equation. Remember the quadratic formula? It's used to find the solutions (or roots) of any quadratic equation in the form ax² + bx + c = 0, and it looks like this:

x = (-b ± √(b² - 4ac)) / 2a

The discriminant is the part under the square root, which is b² - 4ac. This little expression holds the key to understanding whether the roots are real, distinct, or complex.

  • If b² - 4ac > 0, the equation has two distinct real roots. This is what we need to prove in our case.
  • If b² - 4ac = 0, the equation has exactly one real root (a repeated root).
  • If b² - 4ac < 0, the equation has two complex roots (no real roots).

So, our mission is to show that for the equation kx² + 5x - 2k = 0 and for all positive values of k (where k ≠ 0), the discriminant b² - 4ac is always greater than zero. This will confirm that the roots are indeed real and distinct.

Applying the Discriminant to Our Equation

Now, let's identify the coefficients in our equation kx² + 5x - 2k = 0. Comparing this to the general form ax² + bx + c = 0, we can see that:

  • a = k
  • b = 5
  • c = -2k

Next, we'll plug these values into the discriminant formula b² - 4ac:

Discriminant = (5)² - 4(k)(-2k)

Simplify this expression:

Discriminant = 25 + 8k²

Alright, we've got our discriminant! Now, we need to analyze it and show that it's always greater than zero for positive values of k.

Proving the Discriminant is Positive

We've found that the discriminant is 25 + 8k². Let's think about what this expression tells us. We know that k² will always be non-negative because squaring any real number results in a positive number or zero. Since k ≠ 0, then k² must be a positive number.

Multiplying a positive k² by 8 (which is also positive) will always result in a positive number. So, 8k² is always greater than zero for k ≠ 0. Now, we're adding 25 to this positive value. Adding a positive number to another positive number will always result in a positive number.

Therefore, 25 + 8k² is always greater than zero for any positive value of k (where k ≠ 0). This is exactly what we needed to show! We've proven that the discriminant is positive, which means the roots of the equation kx² + 5x - 2k = 0 are real and distinct when k is positive and not equal to zero.

Conclusion

So, there you have it! We've successfully demonstrated that the roots of the quadratic equation kx² + 5x - 2k = 0 are real and distinct for all positive values of k (where k ≠ 0). We did this by understanding the concept of the discriminant, applying it to our specific equation, and then logically showing that the discriminant is always greater than zero under the given conditions. Remember, the discriminant is your friend when it comes to understanding the nature of roots in quadratic equations. Keep practicing, and you'll master these concepts in no time!

Additional Insights and Tips

Let's dig a bit deeper and explore some additional insights and tips that can help you further understand and tackle similar problems.

Visualizing the Quadratic Equation

One helpful way to think about quadratic equations is to visualize them graphically. The equation y = kx² + 5x - 2k represents a parabola. The roots of the equation are the points where the parabola intersects the x-axis (i.e., where y = 0). When we say the roots are real and distinct, it means the parabola crosses the x-axis at two different points.

Since we've proven that the discriminant is positive for positive values of k, we know that the parabola will always intersect the x-axis at two distinct points. This visual representation can provide a more intuitive understanding of what we've proven algebraically.

The Role of the Coefficient k

The coefficient k plays a significant role in the shape and position of the parabola. In the equation kx² + 5x - 2k = 0, k affects the parabola's concavity (whether it opens upwards or downwards) and its vertical stretch. For positive values of k, the parabola opens upwards. The term -2k affects the y-intercept of the parabola. Understanding how these coefficients influence the graph can help you anticipate the nature of the roots.

Generalizing the Approach

The approach we used to solve this problem can be generalized to other quadratic equations. Whenever you need to determine the nature of the roots, the first step is always to calculate the discriminant. By analyzing the sign of the discriminant, you can quickly determine whether the roots are real and distinct, real and repeated, or complex.

Common Mistakes to Avoid

When working with quadratic equations and discriminants, there are a few common mistakes you should be aware of:

  1. Incorrectly identifying coefficients: Make sure you correctly identify a, b, and c from the quadratic equation. A mistake here will lead to an incorrect discriminant.
  2. Miscalculating the discriminant: Double-check your calculations when computing b² - 4ac. Pay attention to signs, especially when dealing with negative coefficients.
  3. Incorrectly interpreting the discriminant: Remember the rules for the discriminant:
    • b² - 4ac > 0: Two distinct real roots
    • b² - 4ac = 0: One real root (repeated)
    • b² - 4ac < 0: Two complex roots
  4. Forgetting the condition k ≠ 0: In our problem, the condition k ≠ 0 is crucial. If k = 0, the equation becomes linear (5x = 0), not quadratic. Always consider any given conditions in the problem statement.

Practice Problems

To solidify your understanding, try applying these concepts to similar problems. Here are a couple of practice problems you can work on:

  1. Show that the roots of the equation x² + (k + 3)x + 3k = 0 are real for all real values of k.
  2. Determine the values of m for which the equation 2x² - mx + 1 = 0 has real and distinct roots.

By working through these problems, you'll gain more confidence in your ability to analyze quadratic equations and their roots.

Real-World Applications

Understanding quadratic equations and their roots is not just an abstract mathematical exercise. Quadratic equations have numerous applications in real-world scenarios, including:

  • Physics: Projectile motion, where the height of a projectile can be modeled by a quadratic equation.
  • Engineering: Designing structures and systems, where quadratic equations can describe stress and strain.
  • Economics: Modeling supply and demand curves, where the equilibrium point can be found by solving a quadratic equation.
  • Computer Graphics: Creating curves and surfaces, where quadratic equations are used to define parabolas and other shapes.

By mastering the concepts related to quadratic equations, you'll be better equipped to tackle a wide range of problems in various fields.

Further Exploration

If you're interested in delving deeper into this topic, here are some areas you might want to explore:

  • Complex Roots: Learn more about complex numbers and how they arise as roots of quadratic equations.
  • Vieta's Formulas: Explore Vieta's formulas, which relate the coefficients of a polynomial to the sums and products of its roots.
  • Quadratic Inequalities: Study how to solve inequalities involving quadratic expressions.
  • Higher-Degree Polynomials: Extend your understanding to polynomials of degree three or higher and their roots.

Math is a journey of continuous learning, and there's always more to discover. Keep asking questions, keep exploring, and keep building your knowledge!

So, guys, remember the key takeaways from our discussion. The discriminant is your best friend when determining the nature of roots, positive k values in our equation lead to real and distinct roots, and visualizing parabolas can make the concepts clearer. Keep practicing, and you'll ace those quadratic equation problems in no time! Happy solving!