If x is real, the expression `(x^(2)-bc)/(2x-b-c)` has no real values between b and c.True or False

Answer

To determine whether the statement "If x is real, the expression \((x^2 - bc)/(2x - b - c)\) has no real values between b and c" is true or false, we will analyze the expression step by step.

### Step 1: Define the Expression
Let \( y = \frac{x^2 - bc}{2x - b - c} \).

### Step 2: Rearranging the Expression
To analyze the values of \( y \), we can rearrange the equation:
\[ y(2x - b - c) = x^2 - bc \]
This can be rewritten as:
\[ x^2 - 2xy + (b + c)y - bc = 0 \]
This is a quadratic equation in terms of \( x \).

### Step 3: Identify Coefficients
The quadratic equation in standard form is:
\[ x^2 - (2y)x + ((b + c)y - bc) = 0 \]
Here, the coefficients are:
- \( A = 1 \)
- \( B = -2y \)
- \( C = (b + c)y - bc \)

### Step 4: Calculate the Discriminant
For the quadratic equation to have real roots, the discriminant must be non-negative:
\[ D = B^2 - 4AC \geq 0 \]
Substituting the coefficients:
\[ D = (-2y)^2 - 4(1)((b + c)y - bc) \geq 0 \]
This simplifies to:
\[ 4y^2 - 4((b + c)y - bc) \geq 0 \]
Dividing through by 4:
\[ y^2 - (b + c)y + bc \geq 0 \]

### Step 5: Factor the Quadratic
The quadratic \( y^2 - (b + c)y + bc \) can be factored as:
\[ (y - b)(y - c) \geq 0 \]

### Step 6: Analyze the Inequality
To find the intervals where this inequality holds, we can analyze the roots \( y = b \) and \( y = c \):
- The expression \( (y - b)(y - c) \) is positive when \( y \) is either less than \( b \) or greater than \( c \).

### Step 7: Conclusion
Thus, the values of \( y \) are in the intervals:
\[ (-\infty, b] \cup [c, \infty) \]
This means that \( y \) does not take any values in the interval \( (b, c) \).

### Final Answer
Therefore, the statement "If x is real, the expression \((x^2 - bc)/(2x - b - c)\) has no real values between b and c" is **True**.

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