Solving -x^2 + 4x ≤ 7 A Step By Step Guide

Alright, guys, let's dive into the fascinating world of quadratic inequalities! If you've ever stumbled upon an equation that looks like

-x^2 + 4x ≤ 7

and felt a bit intimidated, don't worry – you're in the right place. This guide will break down the process step by step, making it super easy to understand. We'll cover everything from the basics of quadratic inequalities to more advanced techniques, ensuring you're well-equipped to tackle any problem that comes your way. So, buckle up, and let's get started!

What are Quadratic Inequalities?

First off, let's define what we're dealing with. Quadratic inequalities are mathematical statements that compare a quadratic expression to another value using inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). A quadratic expression, in its simplest form, looks like ax^2 + bx + c, where a, b, and c are constants, and 'a' isn't zero (because then it wouldn't be quadratic, would it?).

In our example, -x^2 + 4x ≤ 7, we have a classic quadratic inequality. The goal here is to find all the values of 'x' that make this statement true. Think of it like a puzzle – we're trying to find the pieces (values of 'x') that fit perfectly.

Why are these inequalities important? Well, they pop up all over the place in real-world applications, from physics to economics. For instance, you might use them to model the trajectory of a ball, the profit margins of a business, or the range of possible outcomes in a scientific experiment. Understanding how to solve them opens up a whole new world of problem-solving possibilities.

To get started, it's crucial to rearrange the inequality so that one side is zero. This makes our lives much easier when we start looking for solutions. For -x^2 + 4x ≤ 7, we'd subtract 7 from both sides, giving us -x^2 + 4x - 7 ≤ 0. Now, we're in a good position to move forward. Keep this rearranged form in mind as we delve deeper into the methods for solving these inequalities.

Step-by-Step Guide to Solving Quadratic Inequalities

Okay, let's get our hands dirty with the actual solving process. We'll break it down into manageable steps so that it feels less like climbing a mountain and more like a gentle stroll in the park. Trust me; by the end of this section, you'll feel like a quadratic inequality-solving pro!

Step 1: Rearrange the Inequality

As we touched on earlier, the first thing you'll want to do is rearrange the inequality so that one side is zero. This is crucial because it allows us to use some cool techniques based on finding the roots (or zeros) of the quadratic equation. Our example inequality, -x^2 + 4x ≤ 7, becomes -x^2 + 4x - 7 ≤ 0. Remember, the idea here is to get everything on one side, leaving zero on the other.

Why do we do this? Well, think about it: we're trying to find where the quadratic expression is less than or equal to zero. By setting it up this way, we can focus on the sign of the expression (whether it's positive, negative, or zero) and how it changes around the roots.

Step 2: Find the Roots of the Quadratic Equation

Now comes the fun part – finding the roots! The roots of a quadratic equation are the values of 'x' that make the equation equal to zero. In other words, they're the points where the graph of the quadratic equation crosses the x-axis. There are a couple of ways we can find these roots:

1. Factoring: If the quadratic expression can be factored easily, this is often the quickest method. Factoring involves breaking down the expression into two binomials. For instance, if we had x^2 - 5x + 6 = 0, we could factor it as (x - 2)(x - 3) = 0, giving us roots x = 2 and x = 3.

2. Quadratic Formula: When factoring isn't straightforward (which is quite common), we turn to the trusty quadratic formula. For a quadratic equation ax^2 + bx + c = 0, the roots are given by: x = (-b ± √(b^2 - 4ac)) / (2a)

   This formula might look a bit intimidating at first, but it's a powerful tool that works for any quadratic equation. Let's apply it to our example, -x^2 + 4x - 7 = 0. Here, a = -1, b = 4, and c = -7. Plugging these values into the formula, we get:

   x = (-4 ± √(4^2 - 4(-1)(-7))) / (2(-1)) x = (-4 ± √(16 - 28)) / (-2) x = (-4 ± √(-12)) / (-2)

   Uh oh! We've run into a negative number under the square root. This tells us that the roots are complex numbers (involving 'i', the imaginary unit). In the context of real number inequalities, this means the quadratic equation doesn't cross the x-axis.

Step 3: Analyze the Discriminant

Before we move on, let's take a quick detour to understand the discriminant. The discriminant is the part of the quadratic formula under the square root, namely b^2 - 4ac. It gives us a ton of information about the roots without actually calculating them. Here's the lowdown:

• If b^2 - 4ac > 0, the equation has two distinct real roots.
• If b^2 - 4ac = 0, the equation has one real root (a repeated root).
• If b^2 - 4ac < 0, the equation has no real roots (complex roots).

In our example, the discriminant is 4^2 - 4(-1)(-7) = 16 - 28 = -12. Since it's negative, we know there are no real roots. This is super useful because it tells us that the graph of the quadratic expression never crosses the x-axis.

Step 4: Sketch the Graph of the Quadratic Expression

Now, let's visualize what's going on by sketching the graph of the quadratic expression. Since we know it's a parabola (the U-shaped curve that quadratic equations make), we can get a good idea of its shape and position by considering a few key things:

• The Sign of 'a': If 'a' is positive, the parabola opens upwards (like a smiley face). If 'a' is negative, it opens downwards (like a frowny face).
• The Roots: We already found these (or determined that they don't exist in the real number system). The roots are where the parabola intersects the x-axis.
• The Vertex: The vertex is the highest or lowest point on the parabola. Its x-coordinate is given by -b / (2a). We can plug this value back into the quadratic expression to find the y-coordinate.

For our example, -x^2 + 4x - 7, 'a' is -1, so the parabola opens downwards. We know there are no real roots, and the x-coordinate of the vertex is -4 / (2 * -1) = 2. Plugging x = 2 into the expression, we get -2^2 + 4(2) - 7 = -4 + 8 - 7 = -3. So, the vertex is at (2, -3).

Sketching this, we see a downward-opening parabola with its highest point at (2, -3). Since it doesn't cross the x-axis and opens downwards, the entire parabola is below the x-axis.

Step 5: Determine the Solution Set

Finally, we can determine the solution set for our inequality. Remember, we're looking for the values of 'x' that make -x^2 + 4x - 7 ≤ 0. In other words, we want to know where the parabola is less than or equal to zero.

Looking at our sketch, we see that the entire parabola is below the x-axis (i.e., negative). This means that -x^2 + 4x - 7 is always less than zero for any real value of 'x'. Therefore, the solution set is all real numbers. We can write this as:

• x ∈ ℝ (x is an element of the set of real numbers)
• (-∞, ∞) (interval notation)

So, there you have it! We've solved our quadratic inequality. It might seem like a lot of steps, but with practice, it becomes second nature.

Advanced Techniques and Special Cases

Now that we've covered the basics, let's level up our game a bit. Sometimes, quadratic inequalities throw us curveballs, and we need to be prepared with some advanced techniques and knowledge of special cases.

Dealing with Complex Roots

We've already seen an example where the quadratic equation had complex roots. What does this mean for solving the inequality? Well, if the discriminant (b^2 - 4ac) is negative, there are no real roots, and the parabola doesn't cross the x-axis. In these cases, the sign of the quadratic expression is consistent for all real values of 'x'.

So, how do we determine the sign? Just look at the sign of 'a' (the coefficient of x^2). If 'a' is positive, the parabola opens upwards, and the expression is always positive. If 'a' is negative, the parabola opens downwards, and the expression is always negative.

In our example, -x^2 + 4x - 7 ≤ 0, we found complex roots and 'a' was -1 (negative). This meant the expression was always negative, and the solution set was all real numbers.

Inequalities with Strict Inequalities

What happens if we have a strict inequality, like < or >? The process is pretty much the same, but we need to be careful about including the roots in our solution set. If the inequality is strict, we exclude the roots because the expression can't actually equal zero.

For example, if we had -x^2 + 4x - 7 < 0, the solution set would still be all real numbers because the expression is never equal to zero. However, if we had an inequality like x^2 - 4x + 3 > 0, we'd factor it as (x - 1)(x - 3) > 0, find the roots x = 1 and x = 3, and then exclude these points from our solution set. The solution would be x < 1 or x > 3.

Compound Inequalities

Sometimes, you might encounter compound inequalities, which are two inequalities joined by