Finding X-Intercepts Of H(x) = X² + 6x + 5 A Comprehensive Guide

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In the realm of mathematics, quadratic functions play a pivotal role, and understanding their behavior is crucial for various applications. One fundamental aspect of analyzing these functions is identifying their x-intercepts, which are the points where the function's graph intersects the x-axis. These intercepts provide valuable insights into the function's roots and its overall shape. In this comprehensive guide, we will delve into the process of finding the x-intercepts of the quadratic function h(x) = x² + 6x + 5, expressing them as ordered pairs, and determining the total number of x-intercepts this function possesses.

Understanding X-Intercepts and Quadratic Functions

Before we dive into the specifics of our function, let's establish a firm grasp of the core concepts. X-intercepts, also known as roots or zeros of a function, are the points where the function's graph crosses the x-axis. At these points, the y-value of the function is zero. In other words, to find the x-intercepts, we need to solve the equation h(x) = 0.

Quadratic functions are polynomial functions of the second degree, generally expressed in the form f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards depending on the sign of the coefficient a. The x-intercepts of a quadratic function correspond to the points where the parabola intersects the x-axis.

For our specific function, h(x) = x² + 6x + 5, we can identify the coefficients as a = 1, b = 6, and c = 5. Since a = 1 is positive, the parabola opens upwards. Now, let's explore the methods to find the x-intercepts of this function.

Methods for Finding X-Intercepts

There are several methods we can employ to determine the x-intercepts of a quadratic function. The most common techniques include:

  1. Factoring: This method involves expressing the quadratic expression as a product of two linear factors. If we can factor the expression, we can set each factor equal to zero and solve for x to find the x-intercepts.

  2. Quadratic Formula: The quadratic formula is a general solution for finding the roots of any quadratic equation. It states that for an equation ax² + bx + c = 0, the solutions for x are given by:

    x = (-b ± √(b² - 4ac)) / 2a

    This formula provides a direct way to calculate the x-intercepts, regardless of whether the quadratic expression can be factored easily.

  3. Completing the Square: This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be factored easily. While it's a valuable technique, it's often less efficient than factoring or using the quadratic formula for finding x-intercepts.

In our case, let's start by attempting to factor the quadratic expression x² + 6x + 5. If factoring proves challenging, we can always resort to the quadratic formula.

Finding X-Intercepts by Factoring

To factor the expression x² + 6x + 5, we need to find two numbers that add up to 6 (the coefficient of the x term) and multiply to 5 (the constant term). These numbers are 1 and 5. Therefore, we can rewrite the expression as:

x² + 6x + 5 = (x + 1)(x + 5)

Now, to find the x-intercepts, we set h(x) = 0 and solve for x:

(x + 1)(x + 5) = 0

This equation holds true if either (x + 1) = 0 or (x + 5) = 0. Solving these equations, we get:

x + 1 = 0 => x = -1

x + 5 = 0 => x = -5

Thus, the x-intercepts of the function h(x) = x² + 6x + 5 are x = -1 and x = -5. To express these intercepts as ordered pairs, we write them as (-1, 0) and (-5, 0).

Verifying with the Quadratic Formula

For the sake of completeness and to demonstrate the versatility of the quadratic formula, let's use it to find the x-intercepts as well. Recall the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

For our function, h(x) = x² + 6x + 5, we have a = 1, b = 6, and c = 5. Plugging these values into the quadratic formula, we get:

x = (-6 ± √(6² - 4 * 1 * 5)) / (2 * 1)

x = (-6 ± √(36 - 20)) / 2

x = (-6 ± √16) / 2

x = (-6 ± 4) / 2

This gives us two solutions:

x = (-6 + 4) / 2 = -2 / 2 = -1

x = (-6 - 4) / 2 = -10 / 2 = -5

As we can see, the quadratic formula yields the same x-intercepts as factoring: x = -1 and x = -5. This confirms our previous result and reinforces the reliability of both methods.

Determining the Number of X-Intercepts

Now that we have found the x-intercepts, we can answer the question of how many x-intercepts the function h(x) = x² + 6x + 5 has. We identified two distinct x-intercepts: (-1, 0) and (-5, 0). Therefore, the function has two x-intercepts.

In general, a quadratic function can have zero, one, or two x-intercepts. The number of x-intercepts is determined by the discriminant, which is the expression under the square root in the quadratic formula (b² - 4ac).:

  • If b² - 4ac > 0, the function has two distinct x-intercepts.
  • If b² - 4ac = 0, the function has one x-intercept (a repeated root).
  • If b² - 4ac < 0, the function has no real x-intercepts.

For our function, the discriminant is 6² - 4 * 1 * 5 = 16, which is greater than zero. This confirms that the function has two x-intercepts.

Graphing the Function

To visualize our findings, let's sketch the graph of h(x) = x² + 6x + 5. We know the parabola opens upwards, and we have identified the x-intercepts as (-1, 0) and (-5, 0). To get a better sense of the graph, we can also find the vertex, which is the minimum point of the parabola. The x-coordinate of the vertex is given by:

x_vertex = -b / 2a = -6 / (2 * 1) = -3

To find the y-coordinate of the vertex, we plug x_vertex into the function:

h(-3) = (-3)² + 6(-3) + 5 = 9 - 18 + 5 = -4

So, the vertex is at the point (-3, -4). With this information, we can sketch the graph, which will be a parabola passing through the points (-5, 0), (-1, 0), and (-3, -4).

Conclusion

In this comprehensive guide, we have successfully found the x-intercepts of the quadratic function h(x) = x² + 6x + 5. By factoring the quadratic expression and using the quadratic formula, we determined that the x-intercepts are (-1, 0) and (-5, 0). We also confirmed that the function has two x-intercepts, which aligns with the positive value of the discriminant. Furthermore, we discussed the general methods for finding x-intercepts of quadratic functions and the significance of the discriminant in determining the number of x-intercepts. This understanding of x-intercepts is crucial for analyzing quadratic functions and their applications in various fields of mathematics and beyond. We also briefly discussed how to graph the function using the x-intercepts and the vertex, providing a visual representation of our findings.

By mastering these concepts and techniques, you can confidently tackle problems involving quadratic functions and their x-intercepts. Remember, practice is key to solidifying your understanding, so continue to explore various examples and challenges to hone your skills.

Keywords

Quadratic Functions, X-Intercepts, Roots, Zeros, Factoring, Quadratic Formula, Completing the Square, Discriminant, Parabola, Vertex, Ordered Pairs