Question

A quadratic equation is given as $a{{x}^{2}}+bx+c=0$ , where a,b,c are real, has real roots if:
(A). a , b, c are integers
(B). ${{b}^{2}}>3ac$
(C). $ac>0$
(D). $c=0$

Answer 1

Hint: A quadratic equation has 2 roots. Take them as $\alpha$ and $\beta$ .Find the discriminant (D) from this. For different cases where $D>0,D=0$ and $D<0$ , find which of the above gives real root.

Complete step-by-step solution -
We know that a quadratic equation has 2 roots. Now let $\alpha$ and $\beta$ be the roots of the general form of the quadratic equation $a{{x}^{2}}+bx+c=0$ .Then we can write,
$\alpha =\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$ and $\beta =\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$ .
We have been told that a, b and c are real. Hence the nature of the roots $\alpha$ and $\beta$ of equation $a{{x}^{2}}+bx+c=0$ depends on the quantity $\left( {{b}^{2}}-4ac \right)$which is called the discriminant of the quadratic equation $a{{x}^{2}}+bx+c=0$.
We can consider different cases where a, b and c are real and $a\ne 0$ .
Case 1: ${{b}^{2}}-4ac>0$ , discriminant (D) is positive and has real and unequal roots.
Case 2: ${{b}^{2}}-4ac=0$ , D is zero and has real and equal roots.
Case 3: ${{b}^{2}}-4ac<0$ , D is negative and has roots which are imaginary.
From these 3 cases, we can say that ${{b}^{2}}-4ac>0$ , the roots are real and for ${{b}^{2}}-4ac=0$ , the roots are real and equal. Hence if we put $c=0$ , we get ${{b}^{2}}$ which will always be positive.
If we put $ac>0$ , but if it is greater than ${{b}^{2}}$ then the roots will be imaginary.
Thus, a, b, c are real and have real roots if c=0.
$\therefore c=0$ is the correct answer.
$\therefore$ Option (D) is the correct answer.

Note: There are other cases for $\left( {{b}^{2}}-4ac \right)$ when it is a perfect square and not perfect square. But those cases are not relevant here. Thus, when the determinant of the equation should be positive for its roots to be real is ${{b}^{2}}-4ac>0$ . But from the option c=0 is correct. As it reduces determinant ${{b}^{2}}$ which is always positive.