Let a be an odd positive integer. If the roots of a quadratic equation x² − (a − 1)x + 4a + 7 = 0 are integers, then the least value of a is:​

Correct option is C

Given:

The quadratic equation is $$x^2 - (a - 1)x + 4a + 7 = 0$$​​

a is an odd positive integer.

The roots of the quadratic equation are integers.

Solution: 

For the equation; discriminant

D = $$(a - 1)^2 - 4(4a + 7)$$ ​

The discriminant must be a perfect square

Option A: a = 25

D = $$(25 - 1)^2 - 4(4 \times 25 + 7) = 24^2 - 4(100 + 7) = 576 - 428 = 148$$​

Since 148 is not a perfect square, a = 25 does not give integer roots. 

Option B: a = 19

​$$D = (19 - 1)^2 - 4(4 \times 19 + 7) = 18^2 - 4(76 + 7) = 324 - 316 = 8$$​​

Since 8 is not a perfect square, a = 19 does not give integer roots. 

Option C: a = 21

​$$D = (21 - 1)^2 - 4(4 \times 21 + 7) = 20^2 - 4(84 + 7) = 400 - 364 = 36 = 6^2$$​​

Since 36 is a perfect square, 

Therefore, a = 21 satisfies all conditions. 

Option D: a = 23

D = $$(23 - 1)^2 - 4(4 \times 23 + 7) = 22^2 - 4(92 + 7) = 484 - 380 = 104$$​​

Since 104 is not a perfect square, a = 23 does not give integer roots.

Thus, The correct value of a is 21