Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Solve the given quadratic inequality using the Quadratic Formula. $$ 14 x^{2}+11 x-15 \leq 0 $$

Step-by-Step Solution

Answer

The solution is \( [-\frac{3}{2}, \frac{5}{7}] \).

Step 1: Identify the quadratic equation

The inequality given is \( 14x^2 + 11x - 15 \leq 0 \). First, we identify that the related quadratic equation is \( 14x^2 + 11x - 15 = 0 \). This will help us find the roots, which determine the key points for the inequality.

Step 2: Use the Quadratic Formula

The Quadratic Formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 14 \), \( b = 11 \), and \( c = -15 \). We'll plug these values into the formula to find the roots.

Step 3: Calculate the discriminant

The discriminant \( b^2 - 4ac \) is necessary to solve the formula. Calculate it: \( 11^2 - 4 \times 14 \times (-15) = 121 + 840 = 961 \). Since the discriminant is positive, there will be two real roots.

Step 4: Find the roots

Using the positive discriminant value, apply the quadratic formula. Calculate \( x_1 = \frac{-11 + \sqrt{961}}{28} \) and \( x_2 = \frac{-11 - \sqrt{961}}{28} \). \( \sqrt{961} = 31 \), so the roots are \( x_1 = \frac{-11 + 31}{28} = \frac{20}{28} = \frac{5}{7} \) and \( x_2 = \frac{-11 - 31}{28} = \frac{-42}{28} = -\frac{3}{2} \).

Step 5: Analyze intervals defined by the roots

The roots \( x = -\frac{3}{2} \) and \( x = \frac{5}{7} \) divide the number line into three intervals: \( (-\infty, -\frac{3}{2}) \), \( [-\frac{3}{2}, \frac{5}{7}] \), and \( (\frac{5}{7}, \infty) \). We must check which intervals satisfy the inequality \( 14x^2 + 11x - 15 \leq 0 \).

Step 6: Test each interval

Choose a test point from each interval to determine if it satisfies the inequality. For \( (-\infty, -\frac{3}{2}) \), try \( x = -2 \). For \( [-\frac{3}{2}, \frac{5}{7}] \), check \( x = 0 \). For \( (\frac{5}{7}, \infty) \), use \( x = 1 \). Compute the sign of \( 14x^2 + 11x - 15 \) for each point, finding that the middle interval satisfies the inequality, while the others do not.

Step 7: Write the solution set

The inequality is satisfied between the two roots, inclusive. Therefore, the solution to \( 14x^2 + 11x - 15 \leq 0 \) is the interval \( [-\frac{3}{2}, \frac{5}{7}] \). This includes the endpoints because the inequality is "less than or equal to".

Key Concepts

Quadratic Formula

To solve quadratic inequalities like the one given, we often start by finding the roots of the related quadratic equation. This is where the Quadratic Formula is essential. The formula provides a method to find the roots of any quadratic equation of the form \[ ax^2 + bx + c = 0 \].
The Quadratic Formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
Here's how it works:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Plug these values into the quadratic formula.
• Calculate the results to find the roots \(x_1\) and \(x_2\).

The roots obtained are vital for determining where the quadratic expression changes from positive to negative or vice versa on the number line.

Discriminant

A crucial part of the Quadratic Formula is the discriminant, which is the expression under the square root, denoted as \( b^2 - 4ac \). The discriminant tells us the nature of the roots of the quadratic equation.

• If \(\text{discriminant} > 0\), there are two distinct real roots.
• If \(\text{discriminant} = 0\), there is one real root (a repeated root).
• If \(\text{discriminant} < 0\), there are no real roots (the roots are complex).

In our example, the discriminant is 961, which is positive. This indicates the presence of two distinct real roots. Understanding the discriminant helps predict whether the quadratic graph crosses the x-axis, touches it, or stays entirely above or below it.

Solution Set

Once the roots are determined using the Quadratic Formula, the next step is to find the solution set for the quadratic inequality. The roots divide the number line into sections or intervals.

• Identify these intervals based on the roots found.
• Test each interval to determine whether the original inequality holds.

In our problem, we found roots at \(x = -\frac{3}{2}\) and \(x = \frac{5}{7}\). These roots yield the intervals: \( (-\infty, -\frac{3}{2}) \), \( [-\frac{3}{2}, \frac{5}{7}] \), and \( (\frac{5}{7}, \infty) \). By choosing test points from each interval and substituting them back into the original inequality, we determine the interval \( [-\frac{3}{2}, \frac{5}{7}] \) satisfies the inequality. Hence, this is the solution set.

Interval Analysis

Interval analysis is valuable for solving quadratic inequalities. This method involves examining the sign of the quadratic expression within specific intervals to see where the inequality holds. Here's how it's done:

• Choose test points from each identified interval.
• Substitute these points into the quadratic expression.
• Determine the sign of the result. A positive result means the expression is greater than zero, whereas a negative result means it is less.

In our analysis, we tested points in each interval formed by the roots. For example, using \(x = 0\) in the interval \([-\frac{3}{2}, \frac{5}{7}]\), we found the quadratic expression non-positive, confirming this interval as part of the solution set. Such analysis helps in accurately finding where the inequality holds true on the number line.