Problem 19 - Intermediate Algebra

Question

Solve each inequality. $$ 8 x^{2}+22 x+5 \geq 0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{5}{2}] \cup [-\frac{1}{4}, \infty)\).

Step 1: Identify the Quadratic Coefficients

The given inequality is \(8x^2 + 22x + 5 \geq 0\). Here, the coefficients are: \(a = 8\), \(b = 22\), and \(c = 5\). We'll use these in later steps to find the roots of the quadratic equation.

Step 2: Calculate the Discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substituting the values, we find \(\Delta = 22^2 - 4 \times 8 \times 5 = 484 - 160 = 324\). Since the discriminant is positive, there are two distinct real roots.

Step 3: Find the Roots of the Quadratic Equation

Using the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\), substitute the values to find the roots. We have:\[x = \frac{-22 \pm \sqrt{324}}{16}\]\[x = \frac{-22 \pm 18}{16}\]This results in two roots: \(x_1 = \frac{-4}{16} = -\frac{1}{4}\) and \(x_2 = \frac{-40}{16} = -\frac{5}{2}\).

Step 4: Determine the Intervals of Solution

The roots divide the number line into intervals: \((-\infty, -\frac{5}{2})\), \((-\frac{5}{2}, -\frac{1}{4})\), and \((-\frac{1}{4}, \infty)\). Since the inequality is of the form \(\geq\), test points from each interval to determine where the inequality holds.

Step 5: Test Each Interval

For \((-\infty, -\frac{5}{2})\), test \(x = -3\): \(8(-3)^2 + 22(-3) + 5 = 72 - 66 + 5 = 11\). It's positive, so the inequality holds.For \((-\frac{5}{2}, -\frac{1}{4})\), test \(x = -1\): \(8(-1)^2 + 22(-1) + 5 = 8 - 22 + 5 = -9\). It's negative, so the inequality does not hold.For \((-\frac{1}{4}, \infty)\), test \(x = 1\): \(8(1)^2 + 22(1) + 5 = 8 + 22 + 5 = 35\). It's positive, so the inequality holds.

Step 6: Write the Solution Set

From the interval tests, the inequality \(8x^2 + 22x + 5 \geq 0\) holds in intervals \((-\infty, -\frac{5}{2}] \cup [-\frac{1}{4}, \infty)\). Include the roots as they make the expression equal to zero due to the \(\geq 0\) condition.

Key Concepts

Discriminant

The discriminant is a crucial component in solving quadratic equations. It helps determine the nature of the roots. The discriminant is denoted as \(\Delta\) and is calculated using the formula: \(\Delta = b^2 - 4ac\). This formula arises from the general quadratic equation \(ax^2 + bx + c = 0\). Using the coefficients \(a\), \(b\), and \(c\), it provides insight into the roots without solving for them directly.

• If \(\Delta > 0\), the quadratic equation has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, also known as a repeated or double root.
• If \(\Delta < 0\), the equation has two complex roots, which are not real numbers.

In our problem, the discriminant calculated was \(324\), indicating two distinct real roots exist. Understanding this helps us know the equation's graphs will intersect the x-axis at two points.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It states that the roots \(x\) of the equation \(ax^2 + bx + c = 0\) are given by:\[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]This formula is derived from the process of completing the square and can solve any quadratic equation, whether it is factorable or not.
In the provided exercise, substituting the known values into the formula yielded: \[x = \frac{-22 \pm 18}{16}\]This calculation results in two real solutions since the discriminant was positive, showing the actual points where the parabola defined by the quadratic equation crosses the x-axis. It’s vital because these roots define intervals for testing inequalities.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). They are the points where the graph of the quadratic function touches or intersects the x-axis. In our problem, we identified the roots using the quadratic formula, giving:

• \(x_1 = -\frac{1}{4}\)
• \(x_2 = -\frac{5}{2}\)

These roots effectively split the number line into different intervals that are essential for solving inequalities. By understanding where these roots lie, we can determine the nature of the parabola and how it behaves in different sections of the number line. This is because the sign of the quadratic expression changes at these critical points.

Interval Testing

Interval testing is a method used to solve inequalities by determining which intervals on a number line satisfy the inequality. After calculating the roots, the number line is divided into intervals defined by these roots. For the inequality \(8x^2 + 22x + 5 \geq 0\), the intervals taken were:

• \((-\infty, -\frac{5}{2})\)
• \((-\frac{5}{2}, -\frac{1}{4})\)
• \((-\frac{1}{4}, \infty)\)

Testing a sample point from each interval helps determine whether the parabola lies above or below the x-axis in that interval.
In the problem's context:

• For \((-\infty, -\frac{5}{2})\), testing with \(x = -3\), the expression was positive.
• For \((-\frac{5}{2}, -\frac{1}{4})\), testing with \(x = -1\), the expression was negative.
• For \((-\frac{1}{4}, \infty)\), testing with \(x = 1\), the expression was positive.

The solution set was found by combining intervals where the inequality held true. This method is systematic and ensures all parts of the graph are considered.