For what values of x, the expression x² − 5x − 14 is positive?

To determine the values of x for which the expression x² − 5x − 14 is positive, we first need to find the roots of the equation by setting it to zero:

Finding the Roots

We can use the quadratic formula, which is:

x = (-b ± √(b² - 4ac)) / 2a

In our case, a = 1, b = -5, and c = -14. Plugging these values into the formula gives:

Calculating the Discriminant

First, we calculate the discriminant (b² - 4ac):

• b² = (-5)² = 25
• 4ac = 4 × 1 × (-14) = -56
• Discriminant = 25 - (-56) = 25 + 56 = 81

Finding the Roots

Now, substituting back into the quadratic formula:

x = (5 ± √81) / 2

Since √81 = 9, we have:

• x₁ = (5 + 9) / 2 = 14 / 2 = 7
• x₂ = (5 - 9) / 2 = -4 / 2 = -2

Analyzing the Intervals

The roots are x = -2 and x = 7. These roots divide the number line into three intervals:

• Interval 1: (-∞, -2)
• Interval 2: (-2, 7)
• Interval 3: (7, ∞)

Testing the Intervals

We will test a point from each interval to see where the expression is positive:

• For x = -3 (in Interval 1):

x² − 5x − 14 = 9 + 15 - 14 = 10 (positive)

• For x = 0 (in Interval 2):

x² − 5x − 14 = 0 - 0 - 14 = -14 (negative)

• For x = 8 (in Interval 3):

x² − 5x − 14 = 64 - 40 - 14 = 10 (positive)

Conclusion

The expression x² − 5x − 14 is positive in the intervals:

• x < -2
• x > 7

Thus, the solution is:

x ∈ (-∞, -2) ∪ (7, ∞)