Solve the quadratic inequality by sketching the graph of the corresponding quadratic function. 

x squared minus 2 x minus 35 greater than 0x2−2x−35>0

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Part 1

Select the correct answer below​ and, if​ necessary, fill in the answer box to complete your choice.




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.

Solve the quadratic inequality by sketching the graph of the corresponding quadratic function: x^2 - 2x - 35 > 0

Learn how to solve the quadratic inequality x^2 - 2x - 35 > 0 by sketching the graph of the quadratic function. The solution includes factoring, finding the roots, and performing sign tests on intervals. Suitable for Grades 9-11 students studying quadratic equations and inequalities.

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