Find The Range Of 'm' For Two X-Intercepts

Hey math enthusiasts! Let's dive into a cool quadratic equation problem. We're gonna figure out the values of m that make the graph of the equation y = 3x² + 7x + m have two x-intercepts. Essentially, we're asking, "When does this parabola cross the x-axis twice?" This is a classic problem that combines algebra and a bit of geometric intuition. It's a great opportunity to explore the relationship between quadratic equations, their graphs, and the discriminant. So, buckle up, grab your pens, and let's unravel this mathematical mystery together! We'll break down the concepts, ensuring you grasp the core principles. By the end, you'll be a pro at determining the conditions for x-intercepts. The key lies in understanding the discriminant, a powerful tool in quadratic equations. Let's get started and make math fun!

Decoding the X-Intercepts: What Does It All Mean?

First off, what exactly are x-intercepts? Simply put, they are the points where the graph of a function touches or crosses the x-axis. At these points, the value of y is always zero. Think of it this way: the x-axis is like the ground, and the x-intercepts are where our parabola, the graph of y = 3x² + 7x + m, touches that ground. Now, when does this happen? The number of x-intercepts depends on the equation's specific values, particularly the discriminant of the quadratic formula. A quadratic equation can have zero, one, or two x-intercepts. The condition for two x-intercepts means our parabola dips below the x-axis and then rises back above it, crossing the axis at two distinct points. This behavior is directly related to the solutions (or roots) of the quadratic equation. The quadratic equation in the form of ax² + bx + c = 0 can have real and distinct roots, meaning the graph intersects the x-axis at two points. Understanding the nature of these intercepts helps to visualize the function and predict its behavior. The discriminant helps determine the number of real roots, and thus, the number of x-intercepts.

The Discriminant: Our Secret Weapon

To tackle this problem, we'll use a special tool called the discriminant. It's part of the quadratic formula, a handy formula for solving quadratic equations. The discriminant, often represented by the Greek letter delta (Δ), is the expression inside the square root of the quadratic formula: b² - 4ac. In our equation, y = 3x² + 7x + m, a = 3, b = 7, and c = m. The discriminant tells us about the nature of the roots (solutions) of the quadratic equation. If the discriminant is positive (b² - 4ac > 0), the equation has two distinct real roots, meaning our parabola has two x-intercepts. If it's zero (b² - 4ac = 0), there is exactly one real root (a repeated root), and the parabola touches the x-axis at a single point (the vertex). And if the discriminant is negative (b² - 4ac < 0), there are no real roots, so the parabola doesn't intersect the x-axis at all. Understanding the discriminant is therefore critical to solving this problem and many others involving quadratic equations. We’re essentially using the discriminant to figure out the conditions under which the quadratic equation has two real solutions. This in turn will tell us when the graph of the equation has two x-intercepts.

Solving for m: Let's Get to Work!

Now, let's put our knowledge into action. We know that for the graph to have two x-intercepts, the discriminant (b² - 4ac) must be greater than zero. Using our equation y = 3x² + 7x + m, we can identify the coefficients: a = 3, b = 7, and c = m. Substitute these values into the discriminant formula:

b² - 4ac > 0

(7)² - 4 * 3 * m > 0

49 - 12m > 0

Now, let’s solve this inequality for m. First, subtract 49 from both sides:

-12m > -49

Next, divide both sides by -12. Remember, when you divide or multiply an inequality by a negative number, you must flip the inequality sign. Therefore:

m < 49/12

So, for the graph of y = 3x² + 7x + m to have two x-intercepts, m must be less than 49/12. This means any value of m smaller than 49/12 will result in a parabola that crosses the x-axis twice. This inequality gives us the range of m values that satisfy the condition. The calculation of the discriminant, followed by solving the inequality, is the key to finding the answer. Therefore, we have found that the correct answer is option C.

Visualize the Solution

To help visualize this, imagine the parabola shifting up and down depending on the value of m. If m is large (greater than 49/12), the parabola sits entirely above the x-axis, hence no intercepts. As m decreases and gets close to 49/12, the parabola touches the x-axis at a single point. Finally, when m goes below 49/12, the parabola dips below the x-axis and has two intercepts. This visual understanding reinforces the solution. You can also sketch a few parabolas for different m values to further solidify your understanding. Playing with these values helps to intuitively understand the effect of m on the graph's position and the number of x-intercepts.

Conclusion: Mastering Quadratic Equations

Alright, guys! We've successfully navigated the problem and found the range of m values. By understanding the concept of x-intercepts, the discriminant, and how they relate to the quadratic equation, we have effectively solved the problem. Remember, the discriminant is a powerful tool to determine the nature of the roots of a quadratic equation. It directly affects whether the graph intersects the x-axis at zero, one, or two points. Practice similar problems to strengthen your understanding and gain confidence in solving quadratic equations. Next time you encounter a quadratic equation, you'll be able to quickly determine how its graph behaves. Understanding these concepts will help you become well-versed in quadratic equations and their applications. Keep practicing, and you'll find these problems become easier with time! Remember that math is all about understanding the concepts, and the more you practice, the better you will get. Keep up the excellent work, and always strive to understand the underlying principles.