For what value of x is  $\frac{ x^2 + 2 }{ 10x^2 - 13x - 3}$  is undefined?.

Analyze the question: A fraction is undefined when its denominator equals zero. So we solve 10x² − 13x − 3 = 0.

Concept: Factorise the quadratic by finding two numbers whose product is (10)(−3) = −30 and whose sum is −13. Those numbers are −15 and +2.

Solution:

10x² − 13x − 3 = 0

Split the middle term: 10x² − 15x + 2x − 3 = 0

Group: 5x(2x − 3) + 1(2x − 3) = 0

(2x − 3)(5x + 1) = 0

So:

• 2x − 3 = 0 → x = 3/2
• 5x + 1 = 0 → x = −1/5

Check: For x = 3/2: 10(9/4) − 13(3/2) − 3 = 22.5 − 19.5 − 3 = 0 ✓. For x = −1/5: 10(1/25) + 13/5 − 3 = 0.4 + 2.6 − 3 = 0 ✓.

The fraction is undefined at x = −1/5 and x = 3/2, which is option B.

Tip: "Undefined" for a fraction always means the denominator is zero, so ignore the numerator (x² + 2 doesn't matter here). Always check your roots by substituting them back into the denominator.