Solving The Inequality $c^2 \leq 8c + 20$

Welcome to our friendly guide on solving the inequality $c^2 \leq 8c + 20$. Inequalities might sound a bit intimidating, but don't worry, we're going to break it down step-by-step, making it super clear and easy to understand. Think of it like finding a range of numbers that make a certain statement true, rather than just one specific number. We'll be using some basic algebra and a little bit of number line magic to get to the bottom of this. Whether you're a student tackling homework or just curious about math, this article is for you!

Understanding Inequalities

Before we dive into our specific problem, let's chat about what inequalities are. You're probably familiar with the equals sign (=), which means two things are exactly the same. Inequalities are similar, but instead of equality, they deal with comparisons like 'less than' (<), 'greater than' (>), 'less than or equal to' (≤), and 'greater than or equal to' (≥). So, when we're asked to solve the inequality $c^2 \leq 8c + 20$, we're looking for all the possible values of 'c' that make the left side of the expression ($c^2$) less than or equal to the right side ($8c + 20$). It's like asking, "For which values of 'c' is this statement true?" Unlike equations that often give you one or two specific solutions, inequalities usually give you a range or a set of solutions. This means there might be infinitely many numbers that satisfy the condition, and we represent these solutions using interval notation or on a number line. Understanding this concept is key because it shifts our thinking from finding exact points to finding entire regions where a condition holds. It's a fundamental concept in algebra that opens the door to understanding functions, graphing, and much more complex mathematical ideas. So, let's get ready to explore these ranges!

Rearranging the Inequality

The first practical step when solving most inequalities, including our solve the inequality $c^2 \leq 8c + 20$ problem, is to get all the terms on one side. This makes it easier to analyze the expression. We want to set the inequality to be compared against zero. So, let's move the $8c$ and the $20$ from the right side to the left side. Remember, when you move a term across the inequality sign, you change its sign. Therefore, $8c$ becomes $-8c$ and $20$ becomes $-20$. Our inequality now looks like this: $c^2 - 8c - 20 \leq 0$. This rearranged form is crucial because it allows us to treat the expression as a quadratic function, $f(c) = c^2 - 8c - 20$, and we are now looking for the values of 'c' where this function is less than or equal to zero (i.e., where its graph is on or below the x-axis). This transformation is a standard technique in solving quadratic inequalities, as it brings the problem into a format where we can readily find the roots or 'critical points' of the associated quadratic equation. By setting the inequality to zero, we are effectively finding the boundaries where the expression might change its sign. These boundaries are the roots of the quadratic equation $c^2 - 8c - 20 = 0$, and they divide the number line into intervals. We then test these intervals to see where our original inequality holds true. This systematic approach ensures that we cover all possible solutions and don't miss any part of the solution set. It’s a powerful way to simplify complex comparisons into a more manageable form.

Finding the Roots

Now that we have our inequality in the form $c^2 - 8c - 20 \leq 0$, the next logical step is to find the values of 'c' where the expression equals zero. These are the roots of the quadratic equation $c^2 - 8c - 20 = 0$. Why are these roots important? Because they are the points where the expression $c^2 - 8c - 20$ changes from positive to negative, or vice versa. They act as critical boundaries on our number line. To find these roots, we can use the quadratic formula, which is a reliable method for solving any quadratic equation of the form $ax^2 + bx + c = 0$. The quadratic formula is given by: $c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our equation, $c^2 - 8c - 20 = 0$, we have $a = 1$, $b = -8$, and $c = -20$. Plugging these values into the formula, we get: $c = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(-20)}}{2(1)}$. Let's simplify this: $c = \frac{8 \pm \sqrt{64 + 80}}{2}$. This further simplifies to: $c = \frac{8 \pm \sqrt{144}}{2}$. The square root of 144 is 12, so we have: $c = \frac{8 \pm 12}{2}$. Now, we can find the two roots: $c_1 = \frac{8 + 12}{2} = \frac{20}{2} = 10$ and $c_2 = \frac{8 - 12}{2} = \frac{-4}{2} = -2$. So, the roots are $c = 10$ and $c = -2$. These two numbers, -2 and 10, are the critical points that divide the number line into three distinct intervals: $(-\infty, -2)$, $(-2, 10)$, and $(10, \infty)$. These are the potential regions where our inequality $c^2 - 8c - 20 \leq 0$ might hold true. Finding these roots accurately is fundamental, as they define the boundaries of our solution set. It’s like finding the exact spots on a map where you need to check if you’re inside or outside a certain area. The quadratic formula is an indispensable tool in algebra for precisely this purpose, providing a direct way to calculate these pivotal points.

Testing Intervals

With our critical points, $c = -2$ and $c = 10$, identified, we now know that the number line is divided into three intervals: $(-\infty, -2)$, $(-2, 10)$, and $(10, \infty)$. Our goal is to solve the inequality $c^2 \leq 8c + 20$, which is equivalent to $c^2 - 8c - 20 \leq 0$. This means we are looking for the interval(s) where the expression $c^2 - 8c - 20$ is negative or zero. To do this, we need to pick a test value from each interval and substitute it back into the expression $c^2 - 8c - 20$ to see if the result is negative (satisfying the $\leq 0$ condition) or positive. Let's do this systematically:

1. Interval 1: $(-\infty, -2)$. Let's choose a test value, say $c = -3$. Substitute this into the expression: $(-3)^2 - 8(-3) - 20 = 9 + 24 - 20 = 13$. Since $13$ is positive ($> 0$), this interval does not satisfy our inequality.

2. Interval 2: $(-2, 10)$. Let's choose a test value, say $c = 0$. Substitute this into the expression: $(0)^2 - 8(0) - 20 = 0 - 0 - 20 = -20$. Since $-20$ is negative ($\< 0$), this interval does satisfy our inequality.

3. Interval 3: $(10, \infty)$. Let's choose a test value, say $c = 11$. Substitute this into the expression: $(11)^2 - 8(11) - 20 = 121 - 88 - 20 = 13$. Since $13$ is positive ($> 0$), this interval does not satisfy our inequality.

So, the interval $(-2, 10)$ is where our expression is negative. However, remember our original inequality was $c^2 - 8c - 20 \leq 0$. The 'or equal to' part means that the roots themselves are also part of the solution. Since our roots are $c = -2$ and $c = 10$, we need to include these values. Therefore, the solution includes the interval $(-2, 10)$ and the endpoints -2 and 10. This testing of intervals is a core technique in solving inequalities. It allows us to determine the sign of the expression in different regions defined by its roots. By selecting representative values within each region, we can efficiently map out where the inequality's condition is met. It’s a visual and logical way to confirm which parts of the number line are valid solutions.

The Solution

After carefully testing the intervals defined by the roots $c = -2$ and $c = 10$, we found that the expression $c^2 - 8c - 20$ is negative in the interval $(-2, 10)$. Furthermore, because our original inequality is $c^2 - 8c - 20 \leq 0$ (less than or equal to zero), the roots themselves, where the expression equals zero, are also part of the solution. Thus, we need to include $c = -2$ and $c = 10$ in our solution set. Combining the interval where the expression is negative with the roots where it is zero, we get a closed interval. This means the solution includes all numbers from -2 up to and including 10. In mathematical notation, this is written as $[-2, 10]$. This interval notation is a concise way to represent all the possible values of 'c' that satisfy the original inequality. It means that any number 'c' such that $-2 \leq c \leq 10$ will make the statement $c^2 \leq 8c + 20$ true. This represents a continuous range of numbers, highlighting the power of inequalities in defining solution sets that are more than just discrete points. It's a complete picture of all the values that work! To visualize this, imagine a number line. You would mark -2 and 10 with closed circles (because they are included) and shade the entire region between them. Any number you pick within that shaded region, including -2 and 10, will satisfy the inequality. This is the final answer to our problem of solving the inequality $c^2 \leq 8c + 20$.

Conclusion

We've successfully navigated the process to solve the inequality $c^2 \leq 8c + 20$. By rearranging the inequality to $c^2 - 8c - 20 \leq 0$, finding the roots of the corresponding quadratic equation ($c = -2$ and $c = 10$) using the quadratic formula, and then testing intervals, we determined the solution set. The final solution is the closed interval $[-2, 10]$, meaning all values of 'c' between -2 and 10, inclusive, satisfy the original inequality. This systematic approach is applicable to a wide range of quadratic inequalities. Remember, the key steps are to get one side to zero, find the critical points (roots), and test intervals to see where the inequality holds true. Inequalities are a fundamental part of mathematics, essential for understanding functions, graphing, and problem-solving in various fields. If you'd like to explore more about quadratic equations and inequalities, a great resource is the Math is Fun website, which offers clear explanations and examples.