Directions: Read the following quadratic equations carefully and answer the…
2022
Directions: Read the following quadratic equations carefully and answer the question given below.
(i) x · x − 3x − √(4x2) = −6
(ii) y2 − √(81y2) = −4 × 5
(iii) (z2√(625z6) ⁄ 5z3) + (4 × 7) = 39z
(iv) p2 − (3 × 5)p = 7 × −(8)
In which of the following equation/s is the difference between the larger and smaller root equal to one?
- A.
Only (i), (ii) & (iii)
- B.
Only (i), (ii) & (iv)
- C.
Only (ii) & (iv)
- D.
Only (ii), (iii) & (iv)
- E.
Only (i) & (ii)
Attempted by 1 students.
Show answer & explanation
Correct answer: B
Concept
For a quadratic ax2 + bx + c = 0 with roots p and q, the gap between the roots is |p − q| = √(b2 − 4ac) ⁄ |a|. So an equation has a root-difference of exactly 1 precisely when that value equals 1. As is standard for this exam style, each surd over a perfect even power is taken with its positive coefficient on the variable — √(4x2) = 2x, √(81y2) = 9y, √(625z6) = 25z3 — so every relation reduces to an ordinary quadratic. Reduce each to standard form, factorise, then compare the two roots.
Application — simplify and solve each equation
x2 − 3x − 2x = −6 → x2 − 5x + 6 = 0 → (x − 2)(x − 3) = 0. Roots 2 and 3; gap = 3 − 2 = 1.
y2 − 9y = −20 → y2 − 9y + 20 = 0 → (y − 4)(y − 5) = 0. Roots 4 and 5; gap = 5 − 4 = 1.
(z2 · 25z3) ⁄ 5z3 + 28 = 39z → 5z2 + 28 = 39z → 5z2 − 39z + 28 = 0 → (5z − 4)(z − 7) = 0. Roots 0.8 and 7; gap = 7 − 0.8 = 6.2, not 1.
p2 − 15p = −56 → p2 − 15p + 56 = 0 → (p − 7)(p − 8) = 0. Roots 7 and 8; gap = 8 − 7 = 1.
Cross-check
Read the gaps straight off √(b2 − 4ac) ⁄ |a|: the first relation gives √(25 − 24) = 1, the second √(81 − 80) = 1, the fourth √(225 − 224) = 1 — each a gap of 1. The third gives √(1521 − 560) ⁄ 5 = 31 ⁄ 5 = 6.2, far from 1. So exactly three of the four relations have a root-difference of 1.
Result
The equations whose larger and smaller roots differ by 1 are (i), (ii) and (iv).