Find the values of y for which the expression $\frac{y^2 - 9y + 18}{y^2 + 4y - 21}$ is undefined

Analyze the question: A fraction is undefined when its denominator equals zero. So we only need to solve the denominator: y² + 4y − 21 = 0.

Concept: Division by zero is not allowed. The numerator doesn't decide where the expression is undefined; only the denominator does.

Solution:

Factorise y² + 4y − 21. We need two numbers that multiply to −21 and add to +4. These are +7 and −3.

y² + 4y − 21 = (y + 7)(y − 3)

Set this equal to zero:

(y + 7)(y − 3) = 0

y = −7 or y = 3

Check:

• y = 3: 9 + 12 − 21 = 0 ✓
• y = −7: 49 − 28 − 21 = 0 ✓

Side note: The numerator factorises as y² − 9y + 18 = (y − 3)(y − 6). The factor (y − 3) cancels with the denominator, but the original expression is still undefined at y = 3, because the original form has a zero denominator there. This is why 3 stays in the answer.

So the values are y = 3 and y = −7, which matches option C.

Tip: For "undefined" questions, ignore the numerator and set only the denominator to zero. Don't let cancelling common factors make you drop a value, because the original expression is what counts.