Problem 27 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ 2 x^{2}+5 x+2 \leq 0 $$

Step-by-Step Solution

Answer

The solution is \(-2 \leq x \leq -\frac{1}{2}\).

Step 1: Identify the Quadratic Inequality

The given quadratic inequality is \(2x^2 + 5x + 2 \leq 0\). Our goal is to find the values of \(x\) that satisfy this inequality.

Step 2: Find the Roots of the Quadratic Equation

First, we need to find the roots of the equation \(2x^2 + 5x + 2 = 0\). We can use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 2\), \(b = 5\), and \(c = 2\).

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\) to determine the nature of the roots:\[5^2 - 4 \times 2 \times 2 = 25 - 16 = 9.\]Since the discriminant is positive, there are two distinct real roots.

Step 4: Solve for the Roots

Use the quadratic formula to find the roots:\[x = \frac{-5 \pm \sqrt{9}}{4} = \frac{-5 \pm 3}{4}.\]This gives the roots as \(x = -\frac{1}{2}\) and \(x = -2\).

Step 5: Determine the Sign of the Quadratic Expression

The roots divide the number line into three intervals: 1. \((-\infty, -2)\)2. \((-2, -\frac{1}{2})\)3. \((-\frac{1}{2}, \infty)\)Check the sign of the quadratic expression \(2x^2 + 5x + 2\) in each interval by choosing test points.

Step 6: Analyze Each Interval

Choose a test point in each interval to determine the sign:- In \((-\infty, -2)\), choose \(x = -3\): \(2(-3)^2 + 5(-3) + 2 = 18 - 15 + 2 = 5\), which is positive.- In \((-2, -\frac{1}{2})\), choose \(x = -1\): \(2(-1)^2 + 5(-1) + 2 = 2 - 5 + 2 = -1\), which is negative.- In \((-\frac{1}{2}, \infty)\), choose \(x = 0\): \(2(0)^2 + 5(0) + 2 = 2\), which is positive.

Step 7: Write the Solution for the Inequality

The quadratic expression is less than or equal to zero in the interval where it is negative, i.e., \((-2, -\frac{1}{2})\). Include the boundary points because the inequality is \(\leq 0\): \[-2 \leq x \leq -\frac{1}{2}.\]

Key Concepts

Quadratic Equations

Quadratic equations form the backbone of many algebra problems. They are polynomial equations of the form \( ax^2 + bx + c = 0 \). The coefficients \( a \), \( b \), and \( c \) are real numbers, and \( a \) cannot be zero. The graph of a quadratic equation is a parabola. Understanding the shape and position of the parabola gives valuable information about the solutions, or roots, of the equation. To solve a quadratic equation, finding these roots is crucial. The roots determine where the quadratic touches or crosses the x-axis. These points are important because they help solve related inequalities. Quadratic inequalities like \( 2x^2 + 5x + 2 \leq 0 \) differ slightly as they involve inequality signs. Solving them involves determining intervals where the quadratic is below, above, or on the x-axis.

Discriminant

The discriminant is a key concept when working with quadratic equations. It is derived from the quadratic formula and is represented by \( b^2 - 4ac \). Calculating the discriminant gives information about the nature of the roots without solving for them completely.

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root, and the parabola touches the x-axis at this point.
• If the discriminant is negative, the roots are complex, meaning the parabola does not intersect the x-axis.

In the inequality \( 2x^2 + 5x + 2 \leq 0 \), the discriminant is 9, a positive number, indicating two distinct roots. This is essential as it helps form intervals on the number line to test possible solutions to the inequality.

Quadratic Formula

The quadratic formula is an invaluable tool for finding the roots of any quadratic equation. It provides all the solutions based on the coefficients \( a \), \( b \), and \( c \) in the equation \( ax^2 + bx + c = 0 \). The formula is written as: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Using this formula, you can easily calculate the roots, whether they’re real or complex. For the inequality \( 2x^2 + 5x + 2 \leq 0 \), inserting the values \( a = 2 \), \( b = 5 \), and \( c = 2 \) into the formula results in the roots \( x = -\frac{1}{2} \) and \( x = -2 \).Each root is a critical point where the quadratic might change sign. By testing intervals between and beyond these roots, you can determine where the inequality holds true. The quadratic formula not only finds the roots but also assists in the comprehensive analysis of its graph and the inequality itself.