Hey guys! Today, we're diving into the exciting world of quadratic inequalities. We'll specifically tackle the function f(x) = x² + 4x - 32 and figure out when f(x) ≥ 0. It might sound intimidating, but trust me, it’s super manageable once we break it down step by step. So, let's get started and make math a little less scary, shall we?
Understanding Quadratic Inequalities
Before we jump into the specifics of our function, let's get a grip on what quadratic inequalities actually are. In the simplest terms, a quadratic inequality is a mathematical statement that compares a quadratic expression to another value, often zero. Unlike quadratic equations, which seek specific solutions where the expression equals zero, inequalities deal with ranges where the expression is either greater than, less than, greater than or equal to, or less than or equal to a certain value.
Think of it like this: imagine you're trying to figure out when a roller coaster is above a certain height. The height of the coaster can be modeled by a quadratic equation, and you want to know during what intervals of the ride the coaster is at or above a specific level. That’s essentially what we’re doing when we solve quadratic inequalities.
The general form of a quadratic inequality looks something like this:
- ax² + bx + c > 0
- ax² + bx + c < 0
- ax² + bx + c ≥ 0
- ax² + bx + c ≤ 0
Where a, b, and c are constants, and x is the variable we’re solving for. The heart of solving these inequalities lies in understanding the behavior of the quadratic expression and how it relates to the given inequality.
Why are Quadratic Inequalities Important?
You might be wondering, "Okay, this is interesting, but why should I care about quadratic inequalities?" Well, they pop up in all sorts of real-world scenarios! They're incredibly useful in physics, engineering, economics, and even computer science. For instance, engineers use them to design structures that can withstand certain loads, economists use them to model profit and cost functions, and computer scientists use them in optimization algorithms.
Understanding quadratic inequalities isn't just about passing a math test; it’s about developing a powerful problem-solving tool that can be applied in numerous fields. So, stick with me, and let's unlock the secrets of these fascinating mathematical statements!
Step-by-Step Solution for f(x) ≥ 0
Alright, let’s tackle our specific problem: f(x) = x² + 4x - 32 and we want to find when f(x) ≥ 0. This means we need to determine the values of x for which the quadratic expression is either positive or equal to zero. Don't worry, we'll take it one step at a time.
Step 1: Find the Roots of the Quadratic Equation
The first thing we need to do is find the roots of the quadratic equation f(x) = 0. These roots are the points where the parabola intersects the x-axis, and they are crucial for determining the intervals where the inequality holds true. To find the roots, we set the function equal to zero:
x² + 4x - 32 = 0
Now, we can solve this quadratic equation using factoring, the quadratic formula, or completing the square. Factoring is often the quickest method if we can find factors easily. Let's see if we can factor this one. We’re looking for two numbers that multiply to -32 and add up to 4. Those numbers are 8 and -4. So, we can factor the equation as follows:
(x + 8)(x - 4) = 0
Setting each factor equal to zero gives us the roots:
x + 8 = 0 => x = -8 x - 4 = 0 => x = 4
So, the roots of the equation are x = -8 and x = 4. These are the points where the parabola crosses the x-axis.
Step 2: Sketch the Parabola
Now that we have the roots, it’s incredibly helpful to sketch a quick graph of the parabola. This doesn’t need to be a perfect, detailed graph – a rough sketch will do. We know the parabola opens upwards because the coefficient of the x² term is positive (it’s 1 in this case). We also know it crosses the x-axis at x = -8 and x = 4.
Imagine a U-shaped curve (since it opens upwards) crossing the x-axis at -8 and 4. The parabola will be below the x-axis between these two points and above the x-axis outside of these points. This visual representation will help us determine the intervals where f(x) ≥ 0.
Step 3: Determine the Intervals Where f(x) ≥ 0
We want to find the intervals where the function is greater than or equal to zero. Looking at our sketch, we can see that the parabola is above the x-axis (i.e., f(x) > 0) to the left of x = -8 and to the right of x = 4. Since we also want to include the points where f(x) = 0, we include the roots themselves in our solution.
This gives us two intervals:
- x ≤ -8
- x ≥ 4
Step 4: Express the Solution in Interval Notation
Finally, let's express our solution in interval notation. Interval notation is a concise way to represent a set of numbers using intervals. For x ≤ -8, we use the interval (-∞, -8]. The parenthesis on the left indicates that negative infinity is not included (it's a concept, not a number), and the square bracket on the right indicates that -8 is included in the solution.
Similarly, for x ≥ 4, we use the interval [4, ∞). The square bracket on the left indicates that 4 is included, and the parenthesis on the right indicates that positive infinity is not included.
To combine these two intervals, we use the union symbol