Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Solve the given quadratic inequality using the Quadratic Formula. $$ x^{2}-3 x-4 \geq 0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -1] \cup [4, \infty)\).

Step 1: Identify the quadratic equation

The given inequality is \(x^2 - 3x - 4 \geq 0\). To use the Quadratic Formula, we start by identifying the corresponding quadratic equation: \(x^2 - 3x - 4 = 0\).

Step 2: Apply the quadratic formula

The Quadratic Formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), solves \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = -3\), and \(c = -4\). Substitute these into the formula to get: \[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 1 \times (-4)}}{2 \times 1} \].

Step 3: Simplify the expression

Begin by simplifying under the square root: \((-3)^2 - 4 \times 1 \times (-4) = 9 + 16 = 25\). Then solve for \(x\): \[ x = \frac{3 \pm \sqrt{25}}{2} = \frac{3 \pm 5}{2} \].

Step 4: Calculate the critical points

Solve for \(x\) using the simplified expression: \[ x = \frac{3 + 5}{2} = 4 \quad \text{and} \quad x = \frac{3 - 5}{2} = -1 \]. The critical points are \(x = 4\) and \(x = -1\).

Step 5: Test intervals around the critical points

The critical points divide the number line into three intervals: \((-\infty, -1)\), \((-1, 4)\), and \((4, \infty)\). Choose test points (e.g., \(-2\), \(0\), \(5\)) to determine where the inequality holds. Evaluate the original expression \(x^2 - 3x - 4\) at these points.

Step 6: Evaluate the inequality on each interval

For \(x = -2\) (interval \((-\infty, -1)\)): \((-2)^2 - 3(-2) - 4 = 4 + 6 - 4 = 6\) (positive). For \(x = 0\) (interval \((-1, 4)\)): \(0^2 - 3(0) - 4 = -4\) (negative). For \(x = 5\) (interval \((4, \infty)\)): \(5^2 - 3(5) - 4 = 25 - 15 - 4 = 6\) (positive). Thus, \(x^2 - 3x - 4 \geq 0\) is true on \((-\infty, -1]\) and \([4, \infty)\).

Step 7: Verify inclusion of critical points

Substitute \(x = -1\) and \(x = 4\) into the original expression: \( (-1)^2 - 3(-1) - 4 = 0 \) and \( 4^2 - 3(4) - 4 = 0 \). Both produce zero, which satisfies \(\geq 0\). Include both critical points in the solution set.

Key Concepts

Quadratic Formula

The quadratic formula is a tool used to find the roots of a quadratic equation. It's applicable to any quadratic equation in its standard form: \(ax^2 + bx + c = 0\). The formula is written as:

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

It helps us find the values of \(x\) where the equation equals zero. In our exercise, we have \(a = 1\), \(b = -3\), and \(c = -4\). By substituting these values into the quadratic formula, we solve for the roots of the equation \(x^2 - 3x - 4 = 0\). This process is crucial, as these roots—also called critical points—play a significant role in solving quadratic inequalities.

Critical Points

Critical points are the solutions to the quadratic equation derived from the inequality. They are essential in dividing the number line into intervals for testing. By calculating the critical points, we pinpoint where the expression equals zero.
For \(x^2 - 3x - 4 = 0\), applying the quadratic formula gives the critical points \(x = 4\) and \(x = -1\). These points mark the boundaries between intervals where the inequality expression could change sign. Hence, correctly identifying and calculating these critical points is a vital step in the inequality-solving process.

Interval Testing

Once you have the critical points, the next step is to test the intervals that are formed. The line of real numbers is split into segments by the critical points.

• Here, the intervals will be \((-\ infty, -1)\), \((-1, 4)\), and \((4, \infty)\).

Within each interval, choose a test point, substitute it back into the original inequality, and see if it holds:

• In the interval \((-\ infty, -1)\), a test point like \(x = -2\) shows the result is positive, meaning the inequality is satisfied.
• For the interval \((-1, 4)\), use \(x = 0\). This produces a negative result, which does not satisfy the inequality.
• In the interval \((4, \infty)\), using \(x = 5\), the result is positive, so the inequality is satisfied.

Through interval testing, we determine the parts of the number line satisfying the inequality.

Inequality Solution

The solution to a quadratic inequality involves more than finding critical points. It requires determining the intervals where the inequality is true. After identifying these intervals through testing, we need to verify the inclusion of critical points themselves.
Reassess the original inequality at the critical points \(x = -1\) and \(x = 4\). Both provide zero when substituted back into \(x^2 - 3x - 4\), fulfilling the \(\geq 0\) condition, so they should be included in the solution set.
The complete solution, therefore, is the union of intervals satisfying the inequality, including the critical points, formatted as \((-\ infty, -1] \cup [4, \infty)\). These intervals and points together express where the original inequality holds true, providing a comprehensive view of the solution.