Problem 25 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ -x^{2}-x+12 \geq 0 $$

Step-by-Step Solution

Answer

The solution is \([-4, 3]\).

Step 1: Write the Inequality as an Equation

First, convert the inequality into an equation: \(-x^2 - x + 12 = 0\). This will help us find the boundary points that could satisfy the inequality.

Step 2: Solve the Quadratic Equation

Use the quadratic formula to solve \(-x^2 - x + 12 = 0\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = -1\), \(b = -1\), and \(c = 12\). Substitute these values into the formula to find the roots.

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) which is \((-1)^2 - 4(-1)(12) = 1 + 48 = 49\). Since the discriminant is positive, there are two real roots.

Step 4: Find the Roots

For the equation \(-x^2 - x + 12 = 0\), plug the values into the quadratic formula: \(x = \frac{-(-1) \pm \sqrt{49}}{2(-1)} = \frac{1 \pm 7}{-2}\). Thus, the solutions are \(x = -4\) and \(x = 3\).

Step 5: Determine the Intervals

The roots divide the number line into three intervals: \((-\infty, -4)\), \((-4, 3)\), and \((3, \infty)\). We need to determine which intervals satisfy the original inequality \(-x^2 - x + 12 \geq 0\).

Step 6: Test Each Interval

Choose a test point from each interval and substitute it into the inequality. For example, choose \(x = -5\) for \((-\infty, -4)\), \(x = 0\) for \((-4, 3)\), and \(x = 4\) for \((3, \infty)\). By testing: - For \(x = -5\), the result is negative.- For \(x = 0\), \(-0^2 - 0 + 12 = 12 \geq 0\), satisfying the inequality.- For \(x = 4\), the result is negative.

Step 7: Solution Set

The interval \((-4, 3)\) satisfies the inequality. Since the inequality is \(\geq 0\), include the boundary points where the original function equals zero. Thus, the solution is \([-4, 3]\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \). In our exercise, the quadratic equation is \(-x^2 - x + 12 = 0\). Quadratic equations always have the highest exponent of 2 on the variable, such as \(x^2\). This means the graph of a quadratic equation forms a curve called a parabola.

When solving a quadratic equation, the first step is usually to set it equal to zero as we've done here. By doing this, you can find the solutions or 'roots' using methods like factoring, completing the square, or the quadratic formula. In cases where the equation can't be easily factored, the quadratic formula is a powerful tool:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Let's break down the formula:

• \(b\) and \(c\) are coefficients
• \(\pm\) indicates two possible solutions
• \(\sqrt{b^2 - 4ac}\) is the discriminant, telling us the nature of the roots

This tool is widely used because it will find the roots of any quadratic equation, irrespective of their simplicity or complexity.

Roots of a Quadratic Function

The roots of a quadratic function are the values of \(x\) for which the function \(f(x) = ax^2 + bx + c\) equals zero. In simpler terms, these are the points where the parabola crosses the x-axis. For the quadratic equation \(-x^2 - x + 12 = 0\), finding the roots is essentially finding those x-values where the curve hits or crosses the x-axis.

In our problem, we use the quadratic formula to determine these x-intercepts. After calculating the discriminant (which is 49 in our case), we know it's positive, indicating there are two real roots. Solving \(-x^2 - x + 12 = 0\), we found the roots to be \(x = -4\) and \(x = 3\).

Knowing the roots allows us to see how the curve behaves around these points. It splits the number line into sections which we can test to see where our inequality holds true. For quadratics, the portion of the curve between the roots might be entirely above or below the x-axis and gives statuses like positive or negative values ensuring the satisfaction of the inequality within those sections.

Inequality Intervals

When working with quadratic inequalities like \(-x^2 - x + 12 \geq 0\), determining inequality intervals is crucial. The roots we found earlier, \(x = -4\) and \(x = 3\), divide the number line into distinct sections or intervals:

• \((-\infty, -4)\)
• \((-4, 3)\)
• \((3, \infty)\)

For each interval, we need to check if the inequality holds true by choosing a test point. Here's how it works:

• Select a point within each interval.
• Substitute it into the inequality.
• See if the inequality \(\geq 0\) is satisfied.

Upon testing these intervals, only the interval \((-4, 3)\) satisfied the inequality, meaning that for any x-value within this interval, the function is greater than or equal to zero.

Additionally, the inequality is \(\geq\), meaning the boundary points where \(f(x) = 0\) need to be included. Thus, our solution set becomes \([-4, 3]\), representing all the x-values that satisfy the original quadratic inequality.