We are given the quadratic equation:

$x^2 - 3x + b = 0$

The task is to find the range of values of $b$ such that the roots of this equation are between 1 and 2.

Step 1: Using the properties of the roots

Let the roots of the quadratic equation be denoted by $r_1$ and $r_2$. According to the problem, we know:

$1 < r_1, r_2 < 2$

From Vieta's formulas, for the quadratic equation $x^2 + px + q = 0$, the sum and product of the roots are given by:

$r_1 + r_2 = -\frac{-3}{1} = 3 \quad \text{(sum of roots)}$ $r_1 \cdot r_2 = \frac{b}{1} = b \quad \text{(product of roots)}$

So we have the relations:

$r_1 + r_2 = 3$ $r_1 \cdot r_2 = b$

Step 2: Using the root constraints

We are told that the roots must be between 1 and 2, so:

$1 < r_1 < 2 \quad \text{and} \quad 1 < r_2 < 2$

Now, consider the following:

1. The sum of the roots is $r_1 + r_2 = 3$, which implies that one root is less than 2, and the other is greater than 1.
2. To ensure that both roots are between 1 and 2, we need the quadratic expression to evaluate to positive values at $x = 1$ and $x = 2$. That is, we want the values of the quadratic function at these points to be positive:

$f(1) = 1^2 - 3(1) + b = 1 - 3 + b = b - 2 > 0$

$f(2) = 2^2 - 3(2) + b = 4 - 6 + b = b - 2 > 0$

Thus, both conditions imply:

$b - 2 > 0 \quad \text{or} \quad b > 2$

Step 3: Conclusion

For the roots to lie strictly between 1 and 2, the value of $b$ must be greater than 2:

$b > 2$

Recap:

• The quadratic equation $x^2 - 3x + b = 0$ has roots between 1 and 2 if and only if $b > 2$.

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Would you like more details or have any questions on this? Here are 5 related questions:

1. How do you solve a quadratic equation when the roots are given?
2. What does Vieta's formula tell us about the roots of a quadratic equation?
3. Can the roots of a quadratic equation be outside of the given interval? How would that affect the value of $b$?
4. How can we determine the discriminant of a quadratic equation and its relation to the nature of roots?
5. How would the solution change if the roots were constrained to be between 0 and 1?

Tip: When solving for the range of a parameter in a quadratic equation, always check the values of the quadratic expression at the boundary points of the interval for the roots.