Quadratic Equations & Inequalities — Practice

Question 1

x² − 7x + 12 = 0; y² − 9y + 20 = 0. Relation between x and y?
(a) x > y  (b) x < y  (c) x ≤ y  (d) cannot be established

Accepted Answer

x = 3,4; y = 4,5. Largest x (4) = smallest y (4); otherwise y ≥ x, and no x exceeds any y ⇒ x ≤ y.

Answer: (c) x ≤ y.

Question 2

x² + 2x − 15 = 0. The roots are:
(a) 3, 5  (b) −5, 3  (c) −3, 5  (d) −5, −3

Accepted Answer

c = −15 < 0 ⇒ opposite signs. (x+5)(x−3) ⇒ x = −5, 3.

Answer: (b) −5, 3.

Question 3

x² − 5x + 6 = 0; y² − 7y + 10 = 0. Relation?
(a) x > y  (b) x < y  (c) x = y  (d) cannot be established

Accepted Answer

x = 2,3; y = 2,5. Pairs: 2 vs 2 (=), 2 vs 5 (<), 3 vs 2 (>), 3 vs 5 (<). Some x < y and some x > y ⇒ interleaving. Answer: (d) cannot be established.

Question 4

x² − 9x + 20 = 0; y² − 12y + 35 = 0. Relation?
(a) x < y  (b) x > y  (c) x ≤ y  (d) cannot be established

Accepted Answer

x = 4,5; y = 5,7. 4<5, 4<7, 5=5, 5<7 ⇒ no x exceeds any y, equality at 5 ⇒ x ≤ y.

Answer: (c) x ≤ y.

Question 5

For which range is x² − 5x + 6 < 0? (a) x < 2 or x > 3 (b) 2 < x < 3 (c) x > 3 (d) x < 2

Accepted Answer

(x−2)(x−3) < 0 ⇒ product negative between the roots ⇒ 2 < x < 3.

Answer: (b) 2 < x < 3.

Question 6

x² = 16; y² − 9y + 20 = 0. Relation?
(a) x > y  (b) x ≤ y  (c) x ≥ y  (d) cannot be established

Accepted Answer

x = ±4; y = 4,5. x = −4 < both y; x = 4 = 4 and 4 < 5 ⇒ no x exceeds any y, equality possible ⇒ x ≤ y.

Answer: (b) x ≤ y.

Practice — Quadratic Equations & Inequalities