We are tasked with solving the quadratic inequality:

$x^2 - 2x - 35 > 0$

Step 1: Solve the corresponding quadratic equation

First, find the roots of the corresponding equation:

$x^2 - 2x - 35 = 0$

To solve this, we can factor the quadratic:

$x^2 - 2x - 35 = (x - 7)(x + 5) = 0$

So, the roots are:

$x = 7 \quad \text{or} \quad x = -5$

Step 2: Analyze the inequality

The quadratic inequality we need to solve is:

$(x - 7)(x + 5) > 0$

The roots divide the number line into three intervals:

• $(-\infty, -5)$
• $(-5, 7)$
• $(7, \infty)$

Step 3: Test the intervals

We now check the sign of $(x - 7)(x + 5)$ in each of these intervals:

1. For $x \in (-\infty, -5)$: Pick a value like $x = -6$. $(x - 7)(x + 5) = (-6 - 7)(-6 + 5) = (-13)(-1) = 13 > 0$ So, the expression is positive on this interval.

2. For $x \in (-5, 7)$: Pick a value like $x = 0$. $(x - 7)(x + 5) = (0 - 7)(0 + 5) = (-7)(5) = -35 < 0$ So, the expression is negative on this interval.

3. For $x \in (7, \infty)$: Pick a value like $x = 8$. $(x - 7)(x + 5) = (8 - 7)(8 + 5) = (1)(13) = 13 > 0$ So, the expression is positive on this interval.

Step 4: Conclusion

We need the expression to be greater than zero, so the solution is the intervals where the expression is positive:

$x \in (-\infty, -5) \cup (7, \infty)$

Thus, the solution to the inequality is:

$(-\infty, -5) \cup (7, \infty)$

Graphical Interpretation

• The graph of $y = x^2 - 2x - 35$ is a parabola opening upwards (since the coefficient of $x^2$ is positive).
• The roots $x = -5$ and $x = 7$ are the x-intercepts of the parabola.
• The parabola lies above the x-axis for $x \in (-\infty, -5) \cup (7, \infty)$, which corresponds to the solution of the inequality.

Would you like further details on this? Here are some related questions for additional practice:

1. How do you find the vertex of a quadratic function?
2. What does the sign of the leading coefficient tell you about the graph of a quadratic function?
3. How do you solve a quadratic inequality using the sign test?
4. Can you solve a quadratic inequality using the quadratic formula?
5. What happens when the discriminant of a quadratic equation is zero?

Tip: Always check the sign of the quadratic expression in each interval created by the roots to determine where it satisfies the inequality.