Finding X-Intercepts Of G(x) = X^2 - 10x + 24 A Comprehensive Guide

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#X-intercepts, where a function crosses the x-axis, are crucial points that provide insights into the behavior of the function. In this article, we delve into the quadratic function g(x) = x^2 - 10x + 24 to determine where it intersects the x-axis. This exploration will involve understanding the fundamental concepts of quadratic equations, factoring techniques, and the significance of roots or zeros of a function.

Understanding Quadratic Functions and X-Intercepts

A quadratic function, in its standard form, is represented as f(x) = ax^2 + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards (if a > 0) or downwards (if a < 0). The x-intercepts of a quadratic function, also known as the roots or zeros, are the points where the parabola intersects the x-axis. At these points, the value of the function, g(x), is equal to zero. Therefore, to find the x-intercepts, we need to solve the quadratic equation g(x) = 0.

In our case, the given quadratic function is g(x) = x^2 - 10x + 24. To find the x-intercepts, we need to solve the equation x^2 - 10x + 24 = 0. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this instance, factoring is the most straightforward approach.

Factoring the Quadratic Equation

Factoring involves expressing the quadratic expression as a product of two linear factors. To factor x^2 - 10x + 24, we need to find two numbers that multiply to 24 (the constant term) and add up to -10 (the coefficient of the x term). Let's list the factor pairs of 24:

  • 1 and 24
  • 2 and 12
  • 3 and 8
  • 4 and 6

Among these pairs, -4 and -6 satisfy both conditions: (-4) * (-6) = 24 and (-4) + (-6) = -10. Therefore, we can factor the quadratic expression as:

x^2 - 10x + 24 = (x - 4)(x - 6)

Now, to find the x-intercepts, we set the factored expression equal to zero:

(x - 4)(x - 6) = 0

According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Thus, we have two possible cases:

  1. x - 4 = 0 => x = 4
  2. x - 6 = 0 => x = 6

Therefore, the x-intercepts of the function g(x) are x = 4 and x = 6.

Interpreting the X-Intercepts

The x-intercepts, x = 4 and x = 6, represent the points where the parabola intersects the x-axis. These points are (4, 0) and (6, 0). They are also the roots or zeros of the function g(x), which means that when x = 4 or x = 6, the value of the function g(x) is zero. Graphically, these points mark where the parabola crosses the horizontal axis. Understanding these intercepts is crucial for sketching the graph of the parabola and analyzing the function's behavior. They help determine the intervals where the function is positive or negative and provide insights into the function's minimum or maximum value.

The x-intercepts also play a significant role in real-world applications of quadratic functions. For example, in projectile motion, the x-intercepts can represent the points where a projectile lands on the ground. In business and economics, they can represent break-even points where revenue equals cost. In engineering, they can help determine the stability of structures. The ability to find and interpret x-intercepts is therefore a valuable skill in various fields.

Alternative Methods for Finding X-Intercepts

While factoring is an efficient method for finding x-intercepts when the quadratic expression can be easily factored, other methods are available for more complex quadratic equations. One such method is the quadratic formula, which provides a general solution for any quadratic equation of the form ax^2 + bx + c = 0:

x = (-b ± √(b^2 - 4ac)) / 2a

In our case, a = 1, b = -10, and c = 24. Substituting these values into the quadratic formula, we get:

x = (10 ± √((-10)^2 - 4 * 1 * 24)) / (2 * 1) x = (10 ± √(100 - 96)) / 2 x = (10 ± √4) / 2 x = (10 ± 2) / 2

This gives us two solutions:

  • x = (10 + 2) / 2 = 12 / 2 = 6
  • x = (10 - 2) / 2 = 8 / 2 = 4

These solutions match the x-intercepts we found by factoring, confirming the accuracy of our method. Another method is completing the square, which involves manipulating the quadratic equation to form a perfect square trinomial. This method is particularly useful when the quadratic expression is not easily factored and can also be used to derive the quadratic formula.

The Significance of the Discriminant

Before solving a quadratic equation, it's helpful to examine the discriminant, which is the part of the quadratic formula under the square root sign: b^2 - 4ac. The discriminant provides information about the nature and number of real roots (x-intercepts) of the quadratic equation:

  • If the discriminant is positive (b^2 - 4ac > 0), the equation has two distinct real roots, meaning the parabola intersects the x-axis at two points.
  • If the discriminant is zero (b^2 - 4ac = 0), the equation has one real root (a repeated root), meaning the parabola touches the x-axis at one point (the vertex).
  • If the discriminant is negative (b^2 - 4ac < 0), the equation has no real roots, meaning the parabola does not intersect the x-axis.

In our example, the discriminant is (-10)^2 - 4 * 1 * 24 = 100 - 96 = 4, which is positive. This confirms that the equation has two distinct real roots, as we found earlier.

Graphing the Quadratic Function

To visualize the x-intercepts, let's sketch the graph of g(x) = x^2 - 10x + 24. We know that the parabola opens upwards because the coefficient of x^2 is positive (a = 1). We have found the x-intercepts to be (4, 0) and (6, 0). To complete the graph, we need to find the vertex of the parabola. The x-coordinate of the vertex is given by -b / 2a = -(-10) / (2 * 1) = 5. To find the y-coordinate of the vertex, we substitute x = 5 into the function:

g(5) = (5)^2 - 10 * 5 + 24 = 25 - 50 + 24 = -1

Therefore, the vertex of the parabola is (5, -1). We can now sketch the parabola using the x-intercepts and the vertex. The parabola passes through the points (4, 0) and (6, 0), and its lowest point is at the vertex (5, -1). The graph confirms that the function crosses the x-axis at x = 4 and x = 6.

Conclusion: Identifying the X-Intercepts

In summary, to find where the function g(x) = x^2 - 10x + 24 crosses the x-axis, we need to solve the quadratic equation x^2 - 10x + 24 = 0. By factoring the quadratic expression, we found that the equation can be rewritten as (x - 4)(x - 6) = 0. This yields two solutions, x = 4 and x = 6, which correspond to the points (4, 0) and (6, 0) on the coordinate plane. These are the x-intercepts of the function. Understanding how to find and interpret x-intercepts is crucial for analyzing quadratic functions and their applications in various fields. The methods discussed, including factoring, the quadratic formula, and the discriminant, provide a comprehensive toolkit for solving quadratic equations and understanding the behavior of quadratic functions. Remember that the x-intercepts are not just points on a graph; they are significant indicators of the function's characteristics and its relevance in real-world scenarios. Analyzing the discriminant, graphing the function, and understanding the significance of the x-intercepts allows for a deeper comprehension of quadratic functions and their applications.