Finding X-Intercepts Values Of M In Quadratic Equations
Hey guys! Let's dive into a fun math problem that involves finding the values of m for which a quadratic function has two x-intercepts. This is a classic problem in algebra, and we'll break it down step by step so you can totally nail it. We'll not only solve the problem but also discuss the underlying concepts in detail. So, buckle up and let's get started!
Problem Statement
The problem we're tackling today is: For what values of m does the graph of $y = 3x^2 + 7x + m$ have two x-intercepts?
We are given a quadratic equation in the form of $y = 3x^2 + 7x + m$, and our mission, should we choose to accept it (and we do!), is to figure out what values of m will make this parabola cross the x-axis at two distinct points. These points, my friends, are the x-intercepts, also known as the roots or solutions of the quadratic equation. To find these values, we'll journey through the fascinating world of discriminants and quadratic formulas. So, grab your mathematical swords and shields, and let's get going!
Key Concepts: X-Intercepts and Quadratic Equations
Before we jump into the nitty-gritty, let's make sure we're all on the same page with some key concepts. First up, x-intercepts. An x-intercept is simply the point where a graph crosses the x-axis. At this point, the y-value is always zero. So, to find the x-intercepts of any equation, we set y to zero and solve for x. In the context of quadratic equations, the x-intercepts are also known as the roots or solutions of the equation.
Now, let's talk about quadratic equations. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero (otherwise, it would be a linear equation). The graph of a quadratic equation is a parabola, which is a U-shaped curve. This parabola can open upwards (if a > 0) or downwards (if a < 0). The x-intercepts are where this parabola crosses the x-axis. A quadratic equation can have two, one, or no real x-intercepts, depending on the nature of the curve and its position relative to the x-axis. When the parabola intersects the x-axis at two distinct points, the quadratic equation has two distinct real roots. When the vertex of the parabola touches the x-axis (one intersection point), the equation has one real root (or two equal real roots). And when the parabola doesn't intersect the x-axis at all, the equation has no real roots – only complex roots.
In our case, the equation $y = 3x^2 + 7x + m$ is a quadratic equation where a = 3, b = 7, and c = m. We need to find the values of m that will give us two x-intercepts, meaning two real solutions for x when y is zero. So, let's dive deeper into how we can determine this.
The Discriminant: Your Guide to X-Intercepts
The secret weapon in our quest to find the values of m is something called the discriminant. The discriminant is a part of the quadratic formula that tells us about the nature of the roots of a quadratic equation. Remember the quadratic formula? It's the formula we use to solve for x in a quadratic equation of the form $ax^2 + bx + c = 0$:
x = rac{-b ext{\pm\} ext{\sqrt{}} b^2 - 4ac}{2a}
The part under the square root, $b^2 - 4ac$, is the discriminant. We usually denote the discriminant by the Greek letter Δ (delta). So, $ ext{\Delta} = b^2 - 4ac$. The discriminant is super helpful because it tells us how many real roots our quadratic equation has, without us actually having to solve the equation.
Here's the breakdown:
- If $ ext{\Delta} > 0$ (the discriminant is positive), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points, giving us two x-intercepts.
- If $ ext{\Delta} = 0$ (the discriminant is zero), the quadratic equation has one real root (or two equal real roots). This means the parabola touches the x-axis at exactly one point, giving us one x-intercept.
- If $ ext{\Delta} < 0$ (the discriminant is negative), the quadratic equation has no real roots. The roots are complex numbers. This means the parabola does not intersect the x-axis at all, so there are no x-intercepts.
Since we want our quadratic equation to have two x-intercepts, we need the discriminant to be greater than zero. This is the golden rule we'll use to solve our problem. Let's apply this to our equation and see where it takes us!
Applying the Discriminant to Our Problem
Okay, let's bring it all together and apply the discriminant to our specific problem. Our equation is $y = 3x^2 + 7x + m$. Remember, we need to find the values of m for which this equation has two x-intercepts. This means we need the discriminant to be greater than zero.
First, let's identify our a, b, and c values from the equation. Comparing $y = 3x^2 + 7x + m$ to the standard quadratic form $ax^2 + bx + c = 0$, we have:
- a = 3
- b = 7
- c = m
Now, let's plug these values into the discriminant formula, $ ext{\Delta} = b^2 - 4ac$:
ext{\Delta\} = 7^2 - 4(3)(m)
ext{\Delta\} = 49 - 12m
We want two x-intercepts, so we need $ ext{\Delta} > 0$. Let's set up the inequality:
Now, we need to solve this inequality for m. Let's get to it!
Solving the Inequality for m
To solve the inequality $49 - 12m > 0$, we'll follow the usual steps for solving inequalities, keeping in mind that if we multiply or divide by a negative number, we need to flip the inequality sign. Let's walk through it:
First, we want to isolate the term with m. We can do this by subtracting 49 from both sides:
Now, we need to get m by itself. To do this, we'll divide both sides by -12. But remember, we're dividing by a negative number, so we need to flip the inequality sign:
m < rac{-49}{-12}
Simplifying the fraction, we get:
m < rac{49}{12}
So, the values of m for which the graph of $y = 3x^2 + 7x + m$ has two x-intercepts are all the values of m that are less than $ rac{49}{12}$. This is our solution! But let's take it one step further and relate this to the answer choices you often see in multiple-choice questions.
Connecting to Answer Choices
In multiple-choice questions, you'll often see the answer expressed as a decimal or a simplified fraction. Let's convert $ rac{49}{12}$ to a mixed number or a decimal to better understand the range of m values.
rac{49}{12}$ is equal to 4 and $ rac{1}{12}$, or approximately 4.083 as a decimal. So, our inequality is: $m < 4.083
Now, let's look at the answer choices you provided:
A. $m > rac{25}{3}$ B. $m < rac{25}{3}$ C. $m < rac{49}{12}$
We can see that answer choice C directly matches our solution: $m < rac{49}{12}$. But let's also compare it to the other choices to make sure we understand why they're incorrect.
Choice A, $m > rac{25}{3}$, is saying that m is greater than $ rac{25}{3}$, which is approximately 8.33. This is the opposite of what we found – we need m to be less than a certain value, not greater.
Choice B, $m < rac{25}{3}$, is saying that m is less than $ rac{25}{3}$, which is approximately 8.33. While this is in the correct direction (less than), $ rac{25}{3}$ is not the correct boundary. Our boundary is $ rac{49}{12}$, which is much smaller.
Therefore, the correct answer is indeed C. Guys, we nailed it!
Conclusion
So, to wrap it up, we've solved the problem of finding the values of m for which the graph of $y = 3x^2 + 7x + m$ has two x-intercepts. We learned that the key is to use the discriminant, which tells us about the nature of the roots of a quadratic equation. By setting the discriminant greater than zero, we found that m must be less than $ rac{49}{12}$.
Remember, the discriminant is a powerful tool for understanding quadratic equations and their graphs. It helps us determine whether a parabola intersects the x-axis at two points, one point, or not at all. Keep practicing with these concepts, and you'll become a quadratic equation master in no time!
Happy solving, guys! And remember, math can be fun when you break it down step by step. Keep exploring, keep learning, and keep those mathematical gears turning!