In the following, determine the set of values of k for which the given quadratic equation has real roots:
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(i)
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = 9 – 4 × 2 × k
⇒ 9 – 8k ≥ 0
⇒ k ≤ 9/8
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = k2 – 4 × 2 × 3
D ≥ 0
⇒ k2 – 24 ≥ 0
⇒ (k + 2√6)(k – 2√6) ≥ 0
Thus, k ≤ - 2√6 or k ≥ 2√6
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = 25 – 8k
D ≥ 0
⇒ 25 – 8k ≥ 0
⇒ k ≤ 25/8
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = 36 – 4k
⇒ 36 – 4k ≥ 0
⇒ k ≤ 9
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = k2 – 36
⇒ k2 – 36 ≥ 0
⇒ (k – 6)(k + 6) ≥ 0
⇒ k ≥ 6 or k ≤ -6
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = k2 – 4 × 4
⇒ k2 – 16 ≥ 0
⇒ (k + 4)(k – 4) ≥ 0
⇒ k ≥ 4 or k ≤ -4
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = 4 – 12k
⇒ 4 – 12k ≥ 0
⇒ k ≤ 1/3
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = 9k2 – 16
⇒ 9k2 – 16 ≥ 0
⇒ (3k – 4)(3k + 4) ≥ 0
⇒ k ≤ -4/3 or k ≥ (4/3)
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D ≥ 0, roots are real
⇒ D = k2 + 4 × 2 × 4 = k2 + 32
Thus, D is always greater than 0 for all values of k.