Problem 2 - Precalculus: Mathematics for Calculus

Question

The solutions of the inequality \(x^{2}-2 x-3>0\) are the \(x\) -coordinates of the points on the graph of \(y=x^{2}-2 x-3\) that lie ______ the \(x\) -axis.

Step-by-Step Solution

Answer

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Step 1: Rewrite the Inequality

The given inequality is \(x^{2} - 2x - 3 > 0\). This means we want to find the set of \(x\) values for which the quadratic expression \(x^{2} - 2x - 3\) is greater than zero. Additionally, since this expression represents a parabola, points for which this condition holds will lie above the \(x\)-axis.

Step 2: Solve the Quadratic Equation

Before solving the inequality, we need to find the roots of the equation \(x^{2} - 2x - 3 = 0\). Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -2\), and \(c = -3\), we calculate: \[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-3)}}{2\cdot 1} = \frac{2 \pm \sqrt{4 + 12}}{2} = \frac{2 \pm \sqrt{16}}{2} = \frac{2 \pm 4}{2} \]. This gives the roots: \(x = 3\) and \(x = -1\).

Step 3: Test Intervals

The roots \(x = -1\) and \(x = 3\) divide the number line into three intervals: \(x < -1\), \(-1 < x < 3\), and \(x > 3\). For each interval, choose a test point to determine where the quadratic is greater than zero.

Step 4: Check Interval \(x < -1\)

Choose a test point, such as \(x = -2\): \((-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 > 0\). In this interval, the expression is positive.

Step 5: Check Interval \(-1 < x < 3\)

Choose a test point, such as \(x = 0\): \(0^2 - 2(0) - 3 = -3 < 0\). In this interval, the expression is negative.

Step 6: Check Interval \(x > 3\)

Choose a test point, such as \(x = 4\): \(4^2 - 2(4) - 3 = 16 - 8 - 3 = 5 > 0\). In this interval, the expression is positive.

Step 7: Determine Solution Intervals

The expression \(x^{2} - 2x - 3\) is greater than zero for the intervals \(x < -1\) and \(x > 3\). Thus, these are the regions where the points on the graph lie above the \(x\)-axis.

Key Concepts

Parabola and its Graphical Representation

When working with quadratic inequalities, understanding the parabola is crucial. A parabola is the graph of a quadratic function, which takes the shape of a "U" or an inverted "U". Here, the quadratic expression is given as \(y = x^2 - 2x - 3\). In this case, it opens upwards because the coefficient of \(x^2\) is positive.
The graph of this quadratic expression intersects the x-axis at points known as the roots. The section of the parabola that lies above the x-axis corresponds to the solution set of the inequality \(x^2 - 2x - 3 > 0\). Visualizing this can help in understanding which x-values make the inequality true.

Finding the Roots of Quadratic Equations

The roots, or solutions, of a quadratic equation mark the points where the parabola touches or crosses the x-axis. To find these roots for the equation \(x^2 - 2x - 3 = 0\), we use the quadratic formula.

• For the quadratic equation \(ax^2 + bx + c = 0\), the roots are calculated using the formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• In the exercise, substituting \(a = 1\), \(b = -2\), and \(c = -3\) into the formula gives us the roots \(x = 3\) and \(x = -1\).

These roots divide the number line into intervals that we will test to find where the quadratic expression is greater than zero.

Using Test Intervals to Solve Quadratic Inequalities

Once the roots of the quadratic equation are found, they divide the real number line into distinct intervals. Each interval can be tested to determine whether the quadratic expression is positive or negative within it. For \(x^2 - 2x - 3\), the roots \(x = 3\) and \(x = -1\) create three test intervals:

• \(x < -1\)
• \(-1 < x < 3\)
• \(x > 3\)

In each region, we select a test point and plug it into the expression:

• For \(x < -1\), testing with \(x = -2\) shows the expression is positive.
• For \(-1 < x < 3\), testing with \(x = 0\) shows the expression is negative.
• For \(x > 3\), testing with \(x = 4\) shows the expression is positive.

Thus, the solution to the inequality consists of intervals where the expression is positive: \(x < -1\) and \(x > 3\).

Applying the Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations, especially when factoring is cumbersome. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is applied.
In our problem, the calculated discriminant \(b^2 - 4ac\) turned out to be 16, a perfect square, which indicated that the roots are real and distinct. This simplicity allows us to swiftly determine the critical points, \(x = 3\) and \(x = -1\), which are essential in analyzing the inequality.
Using the formula is beneficial because it provides an exact solution for the roots, helping in identifying the intervals where the inequality holds true. Applying the quadratic formula is crucial for interpreting quadratic expressions and solving inequalities effectively.