Finding K For No Zeros In Quadratic Function

by Blender 45 views

Hey guys! Today, we're diving into an interesting problem in mathematics: figuring out when a quadratic function has no real roots (or zeros). Specifically, we're going to explore how to find the values of the parameter 'k' that make a function in the form y = (k-1)x² + 6x - 3 never touch the x-axis. This is a classic problem that combines algebra and a bit of thinking about the nature of quadratic equations. So, let's get started!

Understanding Quadratic Functions and Their Roots

First, let’s recap what we know about quadratic functions. A quadratic function is generally expressed in the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'a' isn't zero. The graph of a quadratic function is a parabola, which is a U-shaped curve. The roots (or zeros) of a quadratic function are the x-values where the parabola intersects the x-axis, meaning f(x) = 0. These roots are crucial because they tell us a lot about the function's behavior. When we talk about a quadratic function having "no zeros," we mean that the parabola never crosses the x-axis.

There are three main scenarios to consider when it comes to the roots of a quadratic function:

  1. Two distinct real roots: The parabola intersects the x-axis at two different points.
  2. One real root (a repeated root): The parabola touches the x-axis at exactly one point (the vertex of the parabola lies on the x-axis).
  3. No real roots: The parabola doesn't intersect the x-axis at all. It either floats entirely above or entirely below the x-axis. This is the scenario we're most interested in today.

To determine which scenario we're dealing with, we often use the discriminant, which is a part of the quadratic formula. So, let's talk about the discriminant and its role in finding roots.

The Discriminant: Your Key to Finding the Roots

The discriminant is a powerful tool for determining the nature of the roots of a quadratic equation without actually solving for them. It's a part of the quadratic formula, specifically the expression under the square root. The quadratic formula itself is used to find the roots of any quadratic equation ax² + bx + c = 0 and is given by:

x = (-b ± √(b² - 4ac)) / (2a)

The discriminant, often denoted by the Greek letter Δ (Delta), is the expression b² - 4ac. It tells us a lot about the roots:

  • If Δ > 0, the equation has two distinct real roots.
  • If Δ = 0, the equation has exactly one real root (a repeated root).
  • If Δ < 0, the equation has no real roots (the roots are complex numbers).

In our case, we want to find the values of 'k' for which the function y = (k-1)x² + 6x - 3 has no zeros. This means we need to find the values of 'k' that make the discriminant less than zero (Δ < 0). So, let’s apply this to our problem.

Applying the Discriminant to Our Problem

Now, let's apply this knowledge to our specific problem: y = (k-1)x² + 6x - 3. Here, we can identify the coefficients as follows:

  • a = (k - 1)
  • b = 6
  • c = -3

We want to find the values of 'k' for which the discriminant is less than zero (Δ < 0). Let's calculate the discriminant:

Δ = b² - 4ac Δ = (6)² - 4(k - 1)(-3) Δ = 36 + 12(k - 1) Δ = 36 + 12k - 12 Δ = 12k + 24

Now, we set the discriminant less than zero:

12k + 24 < 0

Next, we solve this inequality for 'k'.

Solving the Inequality for k

To find the values of 'k' that make the discriminant less than zero, we need to solve the inequality 12k + 24 < 0. Let's go through the steps:

  1. Subtract 24 from both sides: 12k < -24

  2. Divide both sides by 12: k < -2

So, we've found that the function has no real roots when k < -2. However, there's one more important consideration we need to make. Remember that the coefficient 'a' in a quadratic function cannot be zero. If a = 0, the function becomes linear, not quadratic.

In our case, a = (k - 1). So, we need to make sure that (k - 1) ≠ 0, which means k ≠ 1. This is an important condition to keep in mind.

Considering the Condition a ≠ 0

We've determined that k < -2 for the quadratic to have no real roots. However, we also need to ensure that our function remains quadratic, meaning the coefficient of x² (which is k - 1) cannot be zero. So, we have the condition:

k - 1 ≠ 0 k ≠ 1

This condition is already satisfied by our solution k < -2, since all values of k less than -2 are definitely not equal to 1. Therefore, we don't need to make any further adjustments to our solution set.

Final Solution and Interpretation

Alright, guys, after carefully analyzing the discriminant and considering the condition for a quadratic function, we've arrived at our final solution. The function y = (k-1)x² + 6x - 3 has no zeros when k < -2. This means that for any value of 'k' less than -2, the parabola represented by this function will never intersect the x-axis.

Let’s interpret this result graphically. When k < -2, the parabola opens either upwards or downwards, but it’s positioned in such a way that it never touches the x-axis. If you were to graph these functions, you'd see the parabolas floating either entirely above or entirely below the x-axis.

This problem highlights the power of the discriminant in analyzing quadratic functions. By understanding the discriminant, we can quickly determine the nature of the roots without having to solve the quadratic equation explicitly. This is a valuable skill in algebra and calculus, and it's something you'll use again and again.

Wrapping Up and Key Takeaways

So, to wrap things up, let's recap the key steps we took to solve this problem:

  1. Understanding the problem: We needed to find the values of 'k' for which the quadratic function has no real roots.
  2. Using the discriminant: We calculated the discriminant (Δ = b² - 4ac) and set it less than zero (Δ < 0) to ensure no real roots.
  3. Solving the inequality: We solved the inequality for 'k' to find the range of values that satisfy the condition.
  4. Considering a ≠ 0: We checked the condition k - 1 ≠ 0 to ensure the function remained quadratic.
  5. Interpreting the result: We understood what the solution means graphically and in terms of the function's behavior.

I hope this explanation has been helpful and clear! Remember, practice makes perfect, so try applying these steps to similar problems. Understanding the relationship between the discriminant and the nature of roots is crucial for mastering quadratic functions. Keep practicing, and you'll become a pro in no time!