91Ó°ÊÓ

To solve a quadratic inequality, we first need to find the roots by solving the related quadratic equation. Here, we consider the equation: \[3x^2 + 13x + 10 = 0\] To find the solutions, we use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] where \(a = 3\), \(b = 13\), and \(c = 10\). First, calculate the discriminant: \[b^2 - 4ac = 13^2 - 4 \cdot 3 \cdot 10 = 49\] Since the discriminant is positive, the equation has two real roots. Plug in the values: \[x = \frac{-13 \pm \sqrt{49}}{6} = \frac{-13 \pm 7}{6}\] This results in two solutions:

• \(x = -1\)
• \(x = -\frac{10}{3}\)

These roots are important because they help us define the intervals for testing the inequalities.

Understanding the nature of the quadratic functions graphically can vastly simplify determining which intervals satisfy an inequality. The graph of a quadratic function, \[y = 3x^2 + 13x + 10\] is a parabola. Because the coefficient \(a\) is positive, the parabola opens upward.The roots of the quadratic function \(-\frac{10}{3}\) and \(-1\) are the x-intercepts where the graph crosses the x-axis. These intercepts split the x-axis into three intervals:

• \((-\infty, -\frac{10}{3})\)
• \((-\frac{10}{3}, -1)\)
• \((-1, \infty)\)

By placing these points on a graph, we observe that the parabola is below the x-axis between \(-\frac{10}{3}\) and \(-1\), indicating where the quadratic expression holds less than or equal to zero. To understand inequality solutions thoroughly, it helps to visualize and confirm them graphically.

With the quadratic solving and graphical understanding, we now explore inequality intervals. For the quadratic inequality \(3x^2 + 13x + 10 \leq 0\), we test the intervals created by the roots:

• Interval \((-\infty, -\frac{10}{3})\): Test a value like \(x = -4\). The inequality becomes false here because the value is positive.
• Interval \((-\frac{10}{3}, -1)\): Choose \(x = -2\). The result is negative, satisfying \(\leq 0\).
• Interval \((-1, \infty)\): Test \(x = 0\). The outcome is positive, hence false for \(\leq 0\).

For \(3x^2 + 13x + 10 > 0\), the inequalities are true in the outside regions:

• Interval \((-\infty, -\frac{10}{3})\): Tested with \(x = -4\), yielding a positive result.
• Interval \((-\frac{10}{3}, -1)\): With \(x = -2\), results are negative, hence false.
• Interval \((-1, \infty)\): Using \(x = 0\), condition is true because the outcome is positive.

Each interval test confirms where the quadratic expression satisfies the specified inequality, thus providing a full understanding of the solution set.