Simplify: $\frac{x^2 - 5x - 14}{x^2 - 9x + 14}$

Step 1: Analyze the question We need to simplify a rational expression by factorising the numerator and denominator, then cancelling common factors.

Step 2: Concept Factorise each quadratic by finding two numbers whose product is the constant term and whose sum is the coefficient of x.

Step 3: Solution

Numerator: x² - 5x - 14

• We need two numbers with product -14 and sum -5.
• Those are -7 and +2 (since -7 × 2 = -14 and -7 + 2 = -5).
• So x² - 5x - 14 = (x - 7)(x + 2)

Denominator: x² - 9x + 14

• We need two numbers with product +14 and sum -9.
• Those are -7 and -2 (since (-7)(-2) = 14 and -7 + -2 = -9).
• So x² - 9x + 14 = (x - 7)(x - 2)

Put together:

Cancel the common factor (x - 7), where x ≠ 7:

Step 4: Answer The simplified form is (x + 2)/(x - 2), which is option D.

Step 5: Exam tip Always factorise first before cancelling. Never cancel individual terms such as the x² or the numbers directly. Only whole factors can be cancelled. You can also check quickly by substituting a value, e.g. x = 3: the original gives (9 - 15 - 14)/(9 - 27 + 14) = (-20)/(-4) = 5, and option D gives 5/1 = 5. ✓