If a and b are integers in the below mentioned equations then find minimum possible value of a + b^{2}

a^{2}x^{2 }- 4x + 9 < 0

x^{2 }+ bx + 12 ≥ 0

- 38/ 3
- 41/ 3
- 47/ 3
- 51/ 3

Option 1 : 38/ 3

**Calculation:**

a^{2}x^{2} -4x + 9 < 0

Dividing above equation by a^{2}

⇒ x^{2} – (4x/ a^{2}) + (9/ a^{2}) < 0

⇒ (x – (3/ a))^{ 2} + (6x/ a) – (4x/ a^{2}) < 0

⇒ (x – (3/ a))^{ 2} < (4x/ a^{2}) – (6x/ a)

⇒ (4/ a^{2}) – (6/ a) < 0

⇒ 4 – 6a < 0

⇒ 4 > 6a

⇒ a < 2/ 3

x^{2} + bx + 12 ≥ 0

⇒ x^{2} + bx + b^{2} ≥ b^{2} – 12

⇒ (x + b)^{2} ≥ b^{2} – 12

⇒ 12 ≥ b^{2}

⇒ b ≤ 2√ 3

a + b^{2}

⇒ (2/ 3) + (12)

⇒ (2 + 36)/ 3

⇒ 38/ 3

**∴**** required answer is 38/ 3**

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