Question

In the following, determine the set of values of k for which the given quadratic equation has real roots:
(i) 2X²+3X+K=0
(ii) 2X²+KX+3=0
(iii) 2X²-5X-K=0
(iv) KX²+6X+1=0
(v) X²-KX+9=0
(vi) 2X²+KX+2=0
(vii) 3X²+2X+K=0
(viii) 4X²-3KX+1=0
(ix)2X²+KX-4=0

Answer 1

k ≤ 9/8  

Step-by-step explanation:

ax² + bx + c  quadratic equation has real roots

if b² - 4ac ≥ 0

(i) 2X²+3X+K=0

if 3² - 4*2 * k ≥ 0

=> 8k ≤ 9

=> k ≤ 9/8

(ii) 2X²+KX+3=0

=> k² - 4*2 * 3 ≥ 0

=> k² ≥ 24

=> | K |  ≥ 2√6

 k ≤ - 2√6   or  K ≥ 2√6

(iii) 2X²-5X-K=0

(-5)² - 4(2)(-K) ≥ 0

=> 25 + 8K ≥ 0

=> 8K ≥ -25

=> K ≥ -25/8

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