Finding The Values Of 'p' For A Parabola's X-Axis Intersections

Hey guys! Let's dive into a cool math problem. We're talking about parabolas and how they intersect the X-axis. Specifically, we're trying to figure out the values of 'p' that make the graph of the equation $y = px^2 + (p+2)x - p + 4$ intersect the X-axis at two distinct points. This is a classic problem that tests our understanding of quadratic equations and their graphical representations. So, buckle up, and let's break it down step by step. We need to remember some key concepts to solve this problem, including the discriminant of a quadratic equation. This discriminant is our best friend when determining the number of intersections a parabola has with the x-axis.

Understanding the Basics: Quadratic Equations and Parabolas

Alright, first things first. Remember that the general form of a quadratic equation is $ax^2 + bx + c = 0$. And guess what? Our equation, $y = px^2 + (p+2)x - p + 4$, is a quadratic equation in disguise! To find where the graph intersects the X-axis, we need to know where y equals zero. Setting y = 0 allows us to use the quadratic formula to find the roots (or x-intercepts) of the equation. Each root tells us where the parabola crosses the x-axis. In this case, we have $px^2 + (p+2)x - p + 4 = 0$. Now, the coefficients are: $a = p$, $b = p+2$, and $c = -p+4$. These values are super important because they determine the shape and position of the parabola. Since we are looking for two points of intersection, we need to ensure the discriminant is greater than zero.

When we talk about parabolas intersecting the X-axis, we're essentially talking about finding the roots or zeros of the quadratic equation. These roots are the x-values where the parabola touches or crosses the x-axis. A parabola can intersect the x-axis in three ways: it can intersect at two distinct points, touch the x-axis at one point (a repeated root), or not intersect the x-axis at all. The number of intersections is determined by the discriminant, which is the part inside the square root in the quadratic formula: $b^2 - 4ac$. This discriminant is a mathematical tool that gives us a quick way to figure out how many real roots a quadratic equation has, without actually solving the entire equation. So it's very helpful!

The Discriminant: Our Secret Weapon

Now, let's talk about the discriminant in more detail. The discriminant, often denoted as Δ (Delta), is calculated using the formula $\Delta = b^2 - 4ac$. The discriminant is derived directly from the quadratic formula, and it tells us a lot about the nature of the roots of a quadratic equation. If the discriminant is positive ($\Delta > 0$), the quadratic equation has two distinct real roots, which means the parabola intersects the x-axis at two different points. If the discriminant is zero ($\Delta = 0$), the quadratic equation has one real root (a repeated root), and the parabola touches the x-axis at exactly one point (the vertex of the parabola). And if the discriminant is negative ($\Delta < 0$), the quadratic equation has no real roots, meaning the parabola does not intersect the x-axis at all. The discriminant, therefore, is crucial for solving this problem because it lets us know if we'll have two intersection points.

For our specific problem, we want the parabola to intersect the x-axis at two points. This means we need the discriminant to be greater than zero. Let's calculate the discriminant using the coefficients we identified earlier: $a = p$, $b = p+2$, and $c = -p+4$. So, our discriminant formula becomes:

$\Delta = (p+2)^2 - 4(p)(-p+4)$

Let's expand and simplify this:

$\Delta = p^2 + 4p + 4 + 4p^2 - 16p$

$\Delta = 5p^2 - 12p + 4$

To have two intersection points, we need $\Delta > 0$. Therefore, we need to solve the inequality $5p^2 - 12p + 4 > 0$. We're getting closer to solving this problem, and it's all about understanding these steps. By calculating the discriminant and determining when it is greater than zero, we can find the values of 'p' that satisfy the conditions of our problem.

Solving the Inequality: Finding the Range of 'p'

Okay, now we have the inequality $5p^2 - 12p + 4 > 0$. To solve this, we can first find the roots of the quadratic equation $5p^2 - 12p + 4 = 0$ using the quadratic formula, or by factoring if possible. Factoring is usually easier if possible. In this case, we can factor the quadratic expression as follows: $(5p - 2)(p - 2) = 0$. This gives us two possible values for 'p':

$5p - 2 = 0 \implies p = \frac{2}{5}$

$p - 2 = 0 \implies p = 2$

These two values, $\frac{2}{5}$ and 2, are the points where the parabola $5p^2 - 12p + 4$ intersects the p-axis (or the roots of the quadratic equation). Now, let's consider the inequality $5p^2 - 12p + 4 > 0$. Since this is a parabola opening upwards (because the coefficient of $p^2$ is positive), the inequality will be true for values of 'p' less than the smaller root and greater than the larger root. In other words, the solutions to the inequality are $p < \frac{2}{5}$ or $p > 2$. Note that we have to exclude the values of p that make the initial equation undefined or contradict the problem constraints. However, the initial condition is $p eq 0$, so we only need to exclude 0. However, the range of p we calculated above does not include the value 0. Therefore, our final answer must exclude values between the roots $\frac{2}{5}$ and 2.

We're almost there, guys! We've found the critical points where the parabola changes direction, and now we know the intervals where our inequality holds true. This is all that we need to find the correct answer! Remember that this analysis of inequalities is crucial for many mathematical problems, and mastering it helps you with more complex situations.

Determining the Correct Answer

Now, let's look back at the options provided. We're looking for the range of 'p' that makes the parabola intersect the x-axis at two points. We found that this happens when $p < \frac{2}{5}$ or $p > 2$. Looking at the answer choices:

A. $p < 0$ B. $p > 0$ C. $p < \frac{2}{5}$ D. $\frac{2}{5} < p < 2$ E. $p > 3$

The correct answer is the option that matches our solution. Option C, $p < \frac{2}{5}$, is one part of our solution, while option E, $p > 3$, also aligns with our solution set ($p > 2$ includes values greater than 3). Therefore, we should choose options that are a subset of the solution.

By carefully calculating the discriminant and understanding the relationship between the discriminant and the number of x-intercepts, we've successfully solved the problem. The correct answer highlights the intervals where the parabola intersects the x-axis twice. Isn't this fun?

So the correct answer is C and E. Option C, $p < \frac{2}{5}$, is part of the solution.

And option E, $p > 3$, is also a valid range for $p$, because any value of $p$ that satisfies this condition will result in a parabola intersecting the x-axis at two points.

Therefore, the correct answer is E. We made it! Keep practicing, and you'll get better at these problems.