X-Intercepts Of F(x) = X^2 - 2x - 15: A Quick Guide

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X-Intercepts of f(x) = x^2 - 2x - 15: A Quick Guide

Hey guys! Let's dive into finding the x-intercepts of the quadratic function f(x) = x^2 - 2x - 15. X-intercepts, also known as roots or zeros of a function, are the points where the graph of the function crosses the x-axis. At these points, the value of the function, f(x), is equal to zero. So, to find the x-intercepts, we need to solve the equation x^2 - 2x - 15 = 0. This involves a bit of algebraic manipulation, but don't worry, we'll go through it step by step to make sure you get it. Understanding how to find x-intercepts is crucial in various fields, from physics to economics, as it helps in identifying key points in models and equations. It also forms a cornerstone of understanding more complex mathematical concepts. So, let's get started and break down how to solve this quadratic equation and find those x-intercepts!

Finding the X-Intercepts

To find the x-intercepts, we need to solve the equation f(x) = 0. That means we are looking for the values of x that make the function equal to zero. For the given function f(x) = x^2 - 2x - 15, we set up the equation:

x^2 - 2x - 15 = 0

Factoring the Quadratic Equation

The easiest way to solve this quadratic equation is by factoring. We need to find two numbers that multiply to -15 and add up to -2. Those numbers are -5 and +3. Therefore, we can factor the quadratic equation as follows:

(x - 5)(x + 3) = 0

When the product of two factors is zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x:

  • x - 5 = 0 => x = 5
  • x + 3 = 0 => x = -3

Thus, the x-intercepts are x = 5 and x = -3. These are the points where the graph of the function intersects the x-axis. Knowing these points allows us to sketch the graph more accurately and understand the behavior of the function around these critical values. Factoring is a fundamental skill in algebra, and it's super useful not just for finding x-intercepts but also for simplifying expressions and solving more complex equations. Keep practicing, and you'll become a factoring pro in no time!

Identifying the Left-Most and Right-Most X-Intercepts

Now that we have found the x-intercepts, x = 5 and x = -3, we can identify which one is the left-most and which one is the right-most. On the number line, numbers decrease as you move to the left and increase as you move to the right. Therefore:

  • The left-most x-intercept is x = -3
  • The right-most x-intercept is x = 5

So, the coordinates of the x-intercepts are (-3, 0) and (5, 0). These coordinates tell us exactly where the parabola crosses the x-axis. The left-most x-intercept is the point on the left side of the graph, and the right-most x-intercept is the point on the right side. Knowing these points helps us understand the symmetry of the parabola and predict its behavior. In the context of real-world applications, these intercepts might represent break-even points, equilibrium values, or other critical thresholds. Understanding how to identify and interpret these intercepts is super useful for analyzing various phenomena modeled by quadratic functions.

Alternative Methods for Finding X-Intercepts

While factoring is a straightforward method for finding x-intercepts, it's not always applicable, especially when the quadratic equation is not easily factorable. In such cases, alternative methods like the quadratic formula or completing the square can be used.

Using the Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation in the form ax^2 + bx + c = 0. The formula is:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, x^2 - 2x - 15 = 0, we have a = 1, b = -2, and c = -15. Plugging these values into the quadratic formula, we get:

x = (2 ± √((-2)^2 - 4 * 1 * -15)) / (2 * 1) x = (2 ± √(4 + 60)) / 2 x = (2 ± √64) / 2 x = (2 ± 8) / 2

This gives us two solutions:

x = (2 + 8) / 2 = 10 / 2 = 5 x = (2 - 8) / 2 = -6 / 2 = -3

As you can see, we arrive at the same x-intercepts: x = 5 and x = -3. The quadratic formula is a powerful tool because it works for any quadratic equation, regardless of whether it can be easily factored. It's especially useful when dealing with coefficients that are not integers or when the roots are irrational numbers. Knowing how to use the quadratic formula is a fundamental skill in algebra and is essential for solving a wide range of mathematical problems.

Completing the Square

Completing the square is another method for solving quadratic equations. It involves transforming the quadratic equation into a perfect square trinomial. For the equation x^2 - 2x - 15 = 0, we can complete the square as follows:

  1. Move the constant term to the right side of the equation:

    x^2 - 2x = 15

  2. Take half of the coefficient of the x term (-2), square it ((-1)^2 = 1), and add it to both sides of the equation:

    x^2 - 2x + 1 = 15 + 1

  3. Rewrite the left side as a perfect square:

    (x - 1)^2 = 16

  4. Take the square root of both sides:

    x - 1 = ±√16 x - 1 = ±4

  5. Solve for x:

    x = 1 ± 4

This gives us two solutions:

  • x = 1 + 4 = 5
  • x = 1 - 4 = -3

Again, we find the same x-intercepts: x = 5 and x = -3. Completing the square is particularly useful for understanding the vertex form of a quadratic equation, which reveals the vertex of the parabola directly. It also helps in understanding transformations of functions and solving optimization problems. While it might seem a bit more involved than factoring or using the quadratic formula, mastering completing the square provides a deeper understanding of quadratic functions and their properties.

Summary

In summary, the x-intercepts of the function f(x) = x^2 - 2x - 15 are x = -3 and x = 5. Therefore:

  • Left-most x-intercept: (-3, 0)
  • Right-most x-intercept: (5, 0)

We found these x-intercepts by factoring the quadratic equation, and we verified our results using the quadratic formula and completing the square. Understanding how to find x-intercepts is crucial for analyzing quadratic functions and solving related problems. Whether you prefer factoring, using the quadratic formula, or completing the square, having these tools in your arsenal will make you a quadratic equation-solving pro! Keep practicing, and you'll be able to tackle any quadratic equation that comes your way. Remember, math is all about practice, so don't be afraid to make mistakes and learn from them. You got this!