For what values of y is $\frac{y + 2}{8y^2 - 10y  +  3}$ not defined?

Analyze the question: A fraction is not defined when its denominator equals zero. So we solve 8y² − 10y + 3 = 0.

Concept: Factorise the quadratic using the product-sum method. Multiply a × c = 8 × 3 = 24, then find two numbers whose product is 24 and whose sum is −10.

Solution:

The numbers are −4 and −6 (product 24, sum −10).

8y² − 10y + 3 = 0

8y² − 4y − 6y + 3 = 0

4y(2y − 1) − 3(2y − 1) = 0

(2y − 1)(4y − 3) = 0

So:

• 2y − 1 = 0 gives y = 1/2
• 4y − 3 = 0 gives y = 3/4

Check: For y = 3/4: 8(9/16) − 10(3/4) + 3 = 4.5 − 7.5 + 3 = 0 ✓ For y = 1/2: 8(1/4) − 5 + 3 = 2 − 5 + 3 = 0 ✓

The expression is not defined at y = 3/4 and y = 1/2.

Tip: "Not defined" for a fraction always means denominator = 0. Ignore the numerator (y + 2). It only matters if you were asked when the expression equals zero. Also, you can quickly check signs: the middle term is negative and the last term is positive, so both roots must be positive. That eliminates A, B and D immediately.