Question

For any real value of \(x\), if \( \frac{11x^{2}+12x+6}{x^{2}+4x+2} \not\in (a, b) \), then the value for \(x\) for which $$ \frac{11x^{2}+12x+6}{x^{2}+4x+2} = b - a + 3 $$ is: 

• A. \( \frac{3}{4} \)
• B. \( \frac{3}{2} \)
• C. \( 2 \)
• D. \( -\frac{1}{2} \)

Hint:

When working with rational expressions involving quadratics, test specific values of \( x \) to simplify and solve the equation efficiently.

Answer 1

The Correct Option is D

Step 1: Start with the given expression: \[ \frac{11x^2 + 12x + 6}{x^2 + 4x + 2}. \] We need to determine the value of \( x \) such that: \[ \frac{11x^2 + 12x + 6}{x^2 + 4x + 2} = b - a + 3. \] This implies the expression is equal to some constant value. We begin by simplifying the given quadratic expression. 
Step 2: Simplify the expression. The expression is a ratio of two quadratic polynomials, so we need to check for when the numerator and denominator satisfy the condition that results in the value \( b - a + 3 \). For \( x = -\frac{1}{2} \), substitute into the expression: \[ \frac{11(-\frac{1}{2})^2 + 12(-\frac{1}{2}) + 6}{(-\frac{1}{2})^2 + 4(-\frac{1}{2}) + 2}. \] Simplifying the numerator: \[ \frac{11 \times \frac{1}{4} - 6 + 6}{\frac{1}{4} - 2 + 2} = \frac{\frac{11}{4}}{\frac{1}{4}} = 11. \] The denominator simplifies to: \[ \frac{1}{4} - 2 + 2 = \frac{1}{4}. \] Thus, the expression simplifies to: \[ \frac{11}{\frac{1}{4}} = 44. \] Since the condition for \( b - a + 3 \) is satisfied when \( x = -\frac{1}{2} \), the correct answer is \( x = -\frac{1}{2} \).