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To find the value of \( b \) for which the quadratic expression \( x^2 + bx + 1 \) is always positive, we need to ensure that the quadratic has no real roots. This is achieved if the discriminant of the quadratic is less than zero.

The discriminant \(\Delta\) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ \Delta = b^2 - 4ac \]

For the quadratic \( x^2 + bx + 1 \), we have:
- \( a = 1 \)
- \( b = b \)
- \( c = 1 \)

Thus, the discriminant is:
\[ \Delta = b^2 - 4 \cdot 1 \cdot 1 = b^2 - 4 \]

To ensure that \( x^2 + bx + 1 \) is always positive, the discriminant must be less than zero:
\[ b^2 - 4 < 0 \]

Solving this inequality:
\[ b^2 < 4 \]
\[ |b| < 2 \]

So, the values of \( b \) that ensure \( x^2 + bx + 1 \) is always positive are:
\[ -2 < b < 2 \]

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the significance of the discriminant in a quadratic equation?
2. How does the value of \( b \) affect the graph of the quadratic function \( x^2 + bx + 1 \)?
3. What happens to the quadratic \( x^2 + bx + 1 \) if \( b \) is exactly 2 or -2?
4. Can a quadratic equation with real coefficients have no real roots? Under what conditions?
5. How can you determine if a quadratic equation opens upwards or downwards?
6. What are the vertex and axis of symmetry of the quadratic equation \( x^2 + bx + 1 \)?
7. How would you find the minimum value of \( x^2 + bx + 1 \) when \( -2 < b < 2 \)?
8. What is the effect of the constant term in a quadratic equation?

**Tip:** Remember that a quadratic function \( ax^2 + bx + c \) opens upwards if \( a > 0 \) and downwards if \( a < 0 \).

Learn how to ensure that the quadratic expression x^2 + bx + 1 is always positive. Understand the role of the discriminant in quadratic equations and inequalities. Suitable for students in Grades 10-12.

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