Question

For ${ax}^2+bx+c=0$, if $D>0$ and D is square of rational number and a, b, c and d which of the following statements are true

.$i)$ Roots are distinct.

$ii)$ Roots are equal.

$iii)$ Roots are rational

$iv)$ Roots are irrational.

• A. Only statement (i) is true.
• B. Statements (i) and(iii) are true
• C. Only statement (iv) is true
• D. Statements (ii) and (iii) are true

Answer 1

The correct option is B Statements (i) and(iii) are true

Let us recall the general solution, α = (-b-√b²-4ac)/2a and β = (-b+√b²-4ac)/2a

• Case I: D^2 = b² – 4ac > 0

When a, b, and c are real numbers, a ≠ 0 and discriminant is positive, then the roots α and β of the quadratic equation ax² +bx+ c = 0 are real and unequal.

• Case II: b²– 4ac = 0

When a, b, and c are real numbers, a ≠ 0 and discriminant is zero, then the roots α and β of the quadratic equation ax²+ bx + c = 0 are real and equal.

• Case III: b²– 4ac < 0

When a, b, and c are real numbers, a ≠ 0 and discriminant is negative, then the roots α and β of the quadratic equation ax² + bx + c = 0 are unequal and not real. In this case, we say that the roots are imaginary.

• Case IV: b² – 4ac > 0 and perfect square

When a, b, and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax² + bx + c = 0 are real, rational and unequal.

• Case V: b²– 4ac > 0 and not perfect square

When a, b, and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax² + bx + c = 0 are real, irrational and unequal.
Here the roots α and β form a pair of irrational conjugates.

• Case VI: b²– 4ac >0 is perfect square and a or b is irrational

When a, b, and c are real numbers, a ≠ 0 and the discriminant is a perfect square but any one of a or b is irrational then the roots of the quadratic equation ax² + bx + c = 0 are irrational.

Given question is case no (iv) so option B is true.