If x² + 5x + 6 = 0, what is the value of the rational expression 2x / (x² − 7x + 6)?

Introduction / Context:
This algebra problem requires you to use the fact that x satisfies a given quadratic equation to simplify a rational expression involving x. Instead of solving for x and substituting both roots separately, you can use factorisation and algebraic manipulation to show that the expression has the same constant value for both roots.

Given Data / Assumptions:

• x² + 5x + 6 = 0.
• We need to compute 2x / (x² − 7x + 6).
• x is a root of the quadratic equation, so it satisfies the relation x² = −5x − 6.

Concept / Approach:
Key ideas:

• Factorise the quadratic x² + 5x + 6 to find its roots or to express x² in terms of x.
• Simplify the denominator x² − 7x + 6 by factorisation.
• Substitute the values of x corresponding to the roots and verify that the expression gives the same result for both, confirming that it is constant.

Step-by-Step Solution:
Step 1: Factorise the quadratic: x² + 5x + 6 = (x + 2)(x + 3) = 0. Step 2: The roots are x = −2 and x = −3. Step 3: Factorise the denominator: x² − 7x + 6 = (x − 1)(x − 6). Step 4: Evaluate the expression for x = −2: numerator 2x = 2(−2) = −4. Step 5: Denominator for x = −2: (−2)² − 7(−2) + 6 = 4 + 14 + 6 = 24. Step 6: So for x = −2, the expression equals −4 / 24 = −1/6. Step 7: Evaluate the expression for x = −3: numerator 2x = 2(−3) = −6. Step 8: Denominator for x = −3: 9 − 7(−3) + 6 = 9 + 21 + 6 = 36, so the expression is −6 / 36 = −1/6 again. Step 9: Since both roots give the same value, the expression equals −1/6 whenever x satisfies the quadratic.
Verification / Alternative check:
Step 1: Use the relation from the quadratic: x² = −5x − 6. Step 2: Substitute in the denominator: x² − 7x + 6 = (−5x − 6) − 7x + 6 = −12x. Step 3: So the expression becomes 2x / (−12x) = −1/6 for any x that also makes the original quadratic zero and x ≠ 0. Step 4: Neither root is zero, so this simplification is valid, confirming the value −1/6.
Why Other Options Are Wrong:
Options 1/6 and 1/3 have the wrong sign and do not match the computed value. Option −1/3 would appear if the denominator were −6x instead of −12x. Option 0 would require the numerator to be zero, which is not true for either root of the quadratic.
Common Pitfalls:
Some learners incorrectly cancel x from numerator and denominator before using the quadratic relation, which is only safe if x is known to be nonzero and the simplification is valid. Another common error is to substitute only one root and mistakenly assume the expression is equal only for that root, without checking the second root for consistency. Misfactorising either the quadratic or the denominator can lead to incorrect numeric results.
Final Answer:
The value of 2x / (x² − 7x + 6) is −1/6 for any x satisfying x² + 5x + 6 = 0.