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When solving a quadratic inequality like \(2x^2 + 9x - 1 > 0\), one of the most powerful tools in our mathematical toolkit is the quadratic formula. The quadratic formula helps us find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). These roots tell us where the corresponding graph of the quadratic equation intersects the x-axis.

The formula is written as:

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

Here's a breakdown of each symbol in the formula:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation \(ax^2 + bx + c = 0\).
• \(b^2 - 4ac\) is known as the discriminant.
• The symbols \(\pm\) indicate two possible values for \(x\), meaning potentially two real solutions.

In the context of the inequality \(2x^2 + 9x - 1 > 0\), using the quadratic formula helps us find where the expression equals zero, which ultimately helps us determine the solution of the inequality itself.

The discriminant is a key component of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. Given the equation \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated as \(b^2 - 4ac\).

The value of the discriminant provides valuable insights:

• If \(D > 0\), there are two distinct real roots. The quadratic curve crosses the x-axis at two points.
• If \(D = 0\), there is exactly one real root, meaning the graph touches the x-axis at a single point (a perfect square trinomial).
• If \(D < 0\), there are no real roots, indicating that the graph does not intersect the x-axis at all.

In our example, \(2x^2 + 9x - 1 > 0\), the discriminant is \(81 + 8 = 89\), a positive number. This indicates that we have two real solutions, or roots. These roots allow us to determine intervals on the number line to test for solutions to the inequality.

Interval testing is a technique used to determine the solution to a quadratic inequality by evaluating the expression in different intervals derived from its roots. Once we find the roots of the corresponding equation \(2x^2 + 9x - 1 = 0\) using the quadratic formula, we segment the x-axis into several intervals.

The intervals are determined by the roots found using the quadratic formula:

• \((-\infty, x_1)\)
• \((x_1, x_2)\)
• \((x_2, \infty)\)

For each interval, we pick a test point to substitute back into the original inequality \(2x^2 + 9x - 1 > 0\):

• Choose a point like \(-5\) in \((-\infty, x_1)\) to see if the inequality holds.
• Try \(0\) in \((x_1, x_2)\) to check the sign.
• Evaluate \(1\) in \((x_2, \infty)\) for the truth of the inequality.

Based on the test results, you can determine in which intervals the inequality is true. In this example, the inequality holds true in the intervals \((-\infty, -4.805)\) and \((0.105, \infty)\), providing us with the complete solution to the inequality problem.