Directions: Solve the quadratic equations and answer the question that…

Concept

A quadratic in the form ay2 + by + c = 0 is solved by first reducing the given expression to this standard form, then factorising the middle term (split b into two parts whose product is a·c and whose sum is b). The two factors give the two roots. Always simplify every bracket and multiplier before reading off a, b and c.

Application — Equation B

1. Expand the inner bracket times 3: (10y2 − 9y + 2/3) × 3 = 30y2 − 27y + 2. (Note 32 = 9 and (2/3)×3 = 2.)

2. Add the remaining 10y: 30y2 − 27y + 2 + 10y = 30y2 − 17y + 2 = 0. So a = 30, b = −17, c = 2.

3. Split the middle term using a·c = 30 × 2 = 60 and sum = −17: the parts are −12 and −5 (since −12 × −5 = 60 and −12 + −5 = −17).

4. Factorise: 30y2 − 12y − 5y + 2 = 6y(5y − 2) − 1(5y − 2) = (5y − 2)(6y − 1) = 0.

5. Set each factor to zero: 5y − 2 = 0 → y = 2/5; 6y − 1 = 0 → y = 1/6.

Roots of equation B: 2/5 and 1/6.

Cross-check

Sum of roots should equal −b/a = 17/30. Check: 2/5 + 1/6 = 12/30 + 5/30 = 17/30. ✓ Product should equal c/a = 2/30 = 1/15. Check: (2/5)(1/6) = 2/30 = 1/15. ✓ Both match, confirming the roots.