Questions: Solve the following quadratic inequality: x^2 + 2x - 35 < 0 Write your answer in interval notation. Note: Use oo for ∞ and U for union

Solution Steps

To solve the quadratic inequality \(x^2 + 2x - 35 < 0\), we first need to find the roots of the corresponding quadratic equation \(x^2 + 2x - 35 = 0\). These roots will help us determine the intervals to test for the inequality. Once we have the roots, we can test intervals between and beyond these roots to see where the inequality holds true. Finally, we express the solution in interval notation.

To solve the quadratic inequality \(x^2 + 2x - 35 < 0\), we first find the roots of the corresponding equation \(x^2 + 2x - 35 = 0\). The roots are given by:

\[ x = -7 \quad \text{and} \quad x = 5 \]

The roots divide the number line into three intervals: \((-\infty, -7)\), \((-7, 5)\), and \((5, \infty)\). We will test each interval to determine where the inequality \(x^2 + 2x - 35 < 0\) holds true.

1. For the interval \((-\infty, -7)\), choose a test point, e.g., \(x = -8\): \[ (-8)^2 + 2(-8) - 35 = 64 - 16 - 35 = 13 \quad (\text{not less than } 0) \]

2. For the interval \((-7, 5)\), choose a test point, e.g., \(x = 0\): \[ 0^2 + 2(0) - 35 = -35 \quad (\text{less than } 0) \]

3. For the interval \((5, \infty)\), choose a test point, e.g., \(x = 6\): \[ 6^2 + 2(6) - 35 = 36 + 12 - 35 = 13 \quad (\text{not less than } 0) \]

The inequality \(x^2 + 2x - 35 < 0\) holds true in the interval \((-7, 5)\).

Final Answer