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Magazine Chapter 5: Problem 18
Use the graph of the related function of each inequality to write its solutions. $$ x^{2}-4 x-12 \leq 0 $$
Short Answer
Expert verified
The solution to the inequality is \([-2, 6]\).
Step by step solution
01
Identify the Related Function
The related function for the inequality \(x^2 - 4x - 12 \leq 0\) is the quadratic function \(f(x) = x^2 - 4x - 12\). We will use this function to determine the x-values where the inequality holds true.
02
Find the Roots
To find the roots of the quadratic function, set it equal to zero: \(x^2 - 4x - 12 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -12\), calculate the roots.
03
Calculate the Discriminant
Calculate the discriminant \(b^2 - 4ac\) to determine if the quadratic has real roots. Here, \((-4)^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64\), which is positive. This confirms that there are two distinct real roots.
04
Use the Quadratic Formula
Plug in the values into the quadratic formula: \(x = \frac{-(-4) \pm \sqrt{64}}{2 \cdot 1}\) simplifies to \(x = \frac{4 \pm 8}{2}\), yielding the roots \(x = 6\) and \(x = -2\).
05
Analyze the Parabola
The parabola \(f(x) = x^2 - 4x - 12\) opens upwards (since the coefficient of \(x^2\) is positive). The roots \(x = 6\) and \(x = -2\) are where the parabola intersects the x-axis. To solve \(x^2 - 4x - 12 \leq 0\), find the interval where the parabola is below or touches the x-axis.
06
Determine the Solution Interval
Since the parabola opens upwards and the inequality is \(\leq 0\), the function is non-positive between its roots. Therefore, the solution to the inequality \(x^2 - 4x - 12 \leq 0\) is the interval \([-2, 6]\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Related Function
A related function is essentially a key tool we use to simplify the study of quadratic inequalities. When faced with an inequality like \(x^2 - 4x - 12 \leq 0\), our first step is to consider the related quadratic function, \(f(x) = x^2 - 4x - 12\). This function helps us determine the behavior of the inequality across different x-values. You can think of it as the equation we graph or use analytically to explore when and where the inequality holds true.
- This function provides information on where the original inequality is satisfied 鈥� specifically, it will be at or below zero in our example, depending on the graph's shape.
- By finding intervals where the function is zero or negative, we determine the solutions to the inequality.
Roots of a Quadratic
Finding the roots of a quadratic function is a crucial step in solving inequalities like \(x^2 - 4x - 12 \leq 0\). Roots, also known as zeros, are simply the x-values where the function equals zero. Picture these as the points where the graph cuts through the x-axis.
Here's a step-by-step outline to find the roots:
Here's a step-by-step outline to find the roots:
- Set the related quadratic function equal to zero: \(x^2 - 4x - 12 = 0\).
- Use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = -12\).
- Compute the discriminant \(b^2 - 4ac\) to determine the nature of the roots. A positive discriminant indicates that there are two distinct real roots.
- Plug the values into the quadratic formula to find the roots, which in this case are \(x = 6\) and \(x = -2\).
Parabola Analysis
Parabola analysis allows us to make sense of the graph of a quadratic function, such as \(f(x) = x^2 - 4x - 12\). Understanding the parabola's shape helps us interpret the solutions to the inequality. Here are the details you should know:
- The leading coefficient of \(x^2\) is positive, meaning the parabola opens upwards. This is crucial as it determines where the function will be greater or less than zero.
- The x-intercepts of the graph, given by the roots \(x = 6\) and \(x = -2\), are the points where the parabola crosses the x-axis.
- In the context of the inequality, the intervals between and around these roots indicate where the quadratic has negative or non-positive values.
- For our inequality \(x^2 - 4x - 12 \leq 0\), the parabola is non-positive between the roots \(-2\) and \(6\). This interval, therefore, provides the solution to the inequality.
Quadratic Formula
The quadratic formula is a powerhouse tool for finding the roots of a quadratic equation. It's especially helpful when factoring is impractical. For the inequality \(x^2 - 4x - 12 \leq 0\), the quadratic formula helped us find the roots precisely.
Here's how it works:
Here's how it works:
- Identify the coefficients in the quadratic equation: for \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -4\), \(c = -12\).
- Use these values in the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Calculate the discriminant \(b^2 - 4ac\). A discriminant greater than zero means the formula will give two real and distinct solutions.
- Simplify the results: in this example, \(x = \frac{4 \pm 8}{2}\) gives us the roots \(x = 6\) and \(x = -2\).
