Determining Values Of M For Two X-Intercepts In Quadratic Equations

In mathematics, specifically when dealing with quadratic equations, understanding the relationship between the coefficients and the nature of the roots is crucial. This article delves into the process of finding the values of m for which the quadratic equation y = 3x² + 7x + m has two distinct x-intercepts. This exploration involves analyzing the discriminant of the quadratic equation and its implications for the roots. We will discuss the underlying principles, provide a step-by-step solution, and highlight the significance of the discriminant in determining the number of real roots.

To effectively determine the values of m that yield two x-intercepts for the given quadratic equation, it is essential to first grasp the fundamental concepts of quadratic equations and their graphical representations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic equation is a parabola, a U-shaped curve that opens upwards if a > 0 and downwards if a < 0.

X-intercepts, also known as roots or zeros, are the points where the parabola intersects the x-axis. These are the values of x for which y = 0. A quadratic equation can have two, one, or no real x-intercepts, depending on the nature of its discriminant. The discriminant, denoted as Δ (delta), is a crucial component in the quadratic formula and is given by the expression Δ = b² - 4ac. The discriminant provides valuable information about the roots of the quadratic equation:

1. If Δ > 0, the equation has two distinct real roots, meaning the parabola intersects the x-axis at two different points.
2. If Δ = 0, the equation has one real root (a repeated root), meaning the parabola touches the x-axis at one point.
3. If Δ < 0, the equation has no real roots, meaning the parabola does not intersect the x-axis.

In the context of our given equation, y = 3x² + 7x + m, the coefficients are a = 3, b = 7, and c = m. To find the values of m for which the graph has two x-intercepts, we need to ensure that the discriminant is greater than zero. This condition guarantees that the quadratic equation has two distinct real roots, corresponding to two points where the parabola intersects the x-axis. The position of the parabola, and therefore the number of x-intercepts, is significantly influenced by the value of m. By setting up the inequality b² - 4ac > 0 and solving for m, we can determine the range of values for m that satisfy the condition of having two x-intercepts. This understanding is pivotal in analyzing and solving quadratic equations, making it a fundamental concept in algebra.

In our quest to determine the values of m for which the quadratic equation y = 3x² + 7x + m has two x-intercepts, we turn our attention to the discriminant. As previously discussed, the discriminant, represented as Δ, is a critical component in understanding the nature of the roots of a quadratic equation. It is defined by the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation in the standard form ax² + bx + c = 0.

For the given equation, y = 3x² + 7x + m, we can identify the coefficients as follows: a = 3, b = 7, and c = m. Substituting these values into the discriminant formula, we get:

Δ = 7² - 4(3)(m)

Simplifying this expression, we have:

Δ = 49 - 12m

The condition for the quadratic equation to have two distinct x-intercepts is that the discriminant must be greater than zero. This ensures that the equation has two different real roots, which correspond to the two points where the parabola intersects the x-axis. Mathematically, this condition is expressed as:

49 - 12m > 0

To find the values of m that satisfy this inequality, we need to isolate m. This involves a series of algebraic manipulations that will reveal the range of m for which the quadratic equation has two x-intercepts. The process of setting up and analyzing the inequality is a crucial step in solving this problem. By correctly applying the discriminant and the condition for two real roots, we can accurately determine the required values of m. This approach not only provides the solution but also reinforces the understanding of the relationship between the discriminant and the nature of the roots in quadratic equations. Understanding this relationship is fundamental in solving various problems involving quadratic equations and their graphical representations.

Having established the inequality 49 - 12m > 0 as the condition for the quadratic equation y = 3x² + 7x + m to have two x-intercepts, our next step is to solve this inequality for m. This involves a series of algebraic manipulations to isolate m and determine the range of values that satisfy the condition.

Starting with the inequality:

49 - 12m > 0

To isolate the term with m, we can subtract 49 from both sides of the inequality:

-12m > -49

Now, to solve for m, we need to divide both sides of the inequality by -12. It is crucial to remember that when dividing or multiplying an inequality by a negative number, the direction of the inequality sign must be reversed. Therefore, dividing both sides by -12, we get:

m < 49/12

This result indicates that the values of m for which the quadratic equation has two x-intercepts are those less than 49/12. The fraction 49/12 can be expressed as a mixed number or a decimal for better understanding of its value. Converting 49/12 to a mixed number gives us 4 and 1/12, which is approximately 4.083 as a decimal. Thus, the inequality m < 49/12 signifies that m must be less than approximately 4.083 for the quadratic equation to have two distinct x-intercepts.

This solution provides a clear range for the possible values of m. Any value of m less than 49/12 will result in the discriminant being greater than zero, thereby ensuring that the quadratic equation has two real and distinct roots, which graphically represent two x-intercepts. This understanding is vital in various applications of quadratic equations, such as optimization problems, curve fitting, and graphical analysis. The ability to solve inequalities and interpret their results is a fundamental skill in algebra, and this example demonstrates its practical application in the context of quadratic equations.

In conclusion, we have successfully determined the values of m for which the graph of the quadratic equation y = 3x² + 7x + m has two x-intercepts. By understanding the relationship between the discriminant and the nature of the roots, we were able to set up and solve the appropriate inequality. The discriminant, calculated as Δ = b² - 4ac, plays a pivotal role in determining the number of real roots a quadratic equation possesses. For two distinct x-intercepts, the discriminant must be greater than zero.

By applying the discriminant to our given equation, we found that Δ = 49 - 12m. Setting the condition Δ > 0, we derived the inequality 49 - 12m > 0. Solving this inequality for m involved subtracting 49 from both sides and then dividing by -12, remembering to reverse the inequality sign due to the division by a negative number. This process led us to the solution m < 49/12.

Therefore, the graph of y = 3x² + 7x + m has two x-intercepts for all values of m less than 49/12. This solution provides valuable insight into how the constant term m affects the position of the parabola and, consequently, its intersections with the x-axis. The ability to analyze and solve quadratic equations in this manner is a fundamental skill in algebra and has numerous applications in various fields, including physics, engineering, and economics. Mastering these concepts allows for a deeper understanding of mathematical relationships and their graphical representations. The process of determining the values of m not only solves the specific problem but also reinforces the importance of algebraic manipulation, inequality solving, and the interpretation of results within the context of quadratic equations.