Problem 60 - Algebra and Trigonometry

Question

\(59-66\) . Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$ 0.5 x^{2}+0.875 x \leq 0.25 $$

Step-by-Step Solution

Answer

The solutions are the interval [-2, 0.25].

Step 1: Express the Inequality

We start by expressing the given inequality: \[ 0.5 x^2 + 0.875 x \leq 0.25 \]Rearrange it to set it to zero: \[ 0.5 x^2 + 0.875 x - 0.25 \leq 0 \].

Step 2: Multiply Through by 2

To eliminate decimals, multiply every term in the inequality by 2: \[ 2(0.5 x^2) + 2(0.875 x) - 2(0.25) \leq 0 \] which simplifies to:\[ x^2 + 1.75x - 0.5 \leq 0 \].

Step 3: Find the Roots Using the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the roots. Here, \( a = 1 \), \( b = 1.75 \), and \( c = -0.5 \).Calculate the discriminant:\[ \Delta = (1.75)^2 - 4(1)(-0.5) = 3.0625 + 2 = 5.0625 \].Find the square root of the discriminant:\[ \sqrt{5.0625} \approx 2.25 \].Calculate the roots:\[ x_1 = \frac{-1.75 + 2.25}{2} = 0.25 \] and \[ x_2 = \frac{-1.75 - 2.25}{2} = -2 \].

Step 4: Identify the Interval Solution

With roots \( x = 0.25 \) and \( x = -2 \), we know the quadratic will be between these roots for \( x^2 + 1.75x - 0.5 \leq 0 \).This can be verified by checking a test point in the interval, such as \( x = 0 \), which gives: \[ 0^2 + 1.75(0) - 0.5 = -0.5 \leq 0 \].Thus, the solution interval is \([-2, 0.25]\).

Step 5: Draw the Graph to Confirm

Draw a graph of the quadratic function \( y = x^2 + 1.75x - 0.5 \). Observe where it is below the x-axis. The roots \( x = -2 \) and \( x = 0.25 \) should be clear, confirming the interval \([-2, 0.25]\) as the solution.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula helps us to find the roots (or solutions) of these equations by plugging the coefficients \( a \), \( b \), and \( c \) into the formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is particularly useful because it works for any quadratic equation, whether the roots are real or complex. Here is how it works:

• \( -b \): This term changes the sign of \( b \) and is part of finding the coordinate for the axis of symmetry of the parabola.
• \( \sqrt{b^2 - 4ac} \): This is known as the discriminant, and it tells us about the nature and number of the roots.
• \( 2a \): This term in the denominator divides the result into two potential roots due to the \( \pm \) part of the formula, giving us two solutions.

Using the quadratic formula, we can precisely determine the points where the graph of the quadratic equation intersects the x-axis. These intersection points are the solutions, also known as the roots of the equation.

Discriminant

The discriminant is a crucial part of the quadratic formula and it provides valuable insight into the nature of the roots of a quadratic equation. It is represented by the expression \( b^2 - 4ac \). Based on its value, we can determine:

• If \( b^2 - 4ac > 0 \): The quadratic equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \): There is exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \): The equation has two complex roots, which means they do not intersect the x-axis at any point.

For the quadratic inequality in the exercise, the discriminant \( 5.0625 \) was calculated, which is greater than zero. This indicates two distinct real roots \( x_1 = 0.25 \) and \( x_2 = -2 \), validating the graphical solution where the parabola crosses the x-axis at these points.
Hence, understanding the discriminant helps in interpreting the solution set without necessarily graphing the equation first.

Graphical Solution of Inequalities

Graphically solving quadratic inequalities involves sketching the graph of the corresponding quadratic equation and identifying regions where the graph is below or above the x-axis, depending on the inequality sign. Here's how you can do it:

• First, transform the inequality to an equation by setting it as equals \( (ax^2 + bx + c = 0) \) to find the roots.
• Plot these roots on a coordinate grid, which shows where the parabola intersects the x-axis.
• The parabola's direction can be determined from the sign of \( a \). Here, \( a = 1 \), indicating an upward opening parabola.
• Check the portions of the parabola relative to the x-axis. For \( ax^2 + bx + c \leq 0 \), identify the interval where the parabola lies below the x-axis.

In our problem, once the roots \( x = -2 \) and \( x = 0.25 \) were plotted, the solution for the inequality was the interval \([-2, 0.25]\), where the graph is under the x-axis. By inspecting the graph, one can confidently determine this interval as the solution set for the inequality.