If roots of a quadratic equation $ax^{2}+bx+c=0$ are real and equal, then:-

Question

Accepted Answer

The correct option is 1: To solve this problem, we need to recall the concept of the discriminant of a quadratic equation. For a quadratic equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$, the discriminant is given by $D = b^2 - 4ac$. The nature of the roots of the quadratic equation is determined by the value of the discriminant. There are three cases for the nature of the roots based on the discriminant:1. If $D \> 0$ (i.e., $b^2 - 4ac \> 0$), the roots are real and distinct (unequal). 2. If $D = 0$ (i.e., $b^2 - 4ac = 0$), the roots are real and equal. 3. If $D \< 0$ (i.e., $b^2 - 4ac \< 0$), the roots are not real; they are complex (or imaginary).The question states that the roots of the quadratic equation $ax^{2}+bx+c=0$ are real and equal. According to the conditions for the discriminant, this occurs when $D = 0$. Therefore, if the roots are real and equal, then $b^2 - 4ac = 0$. The incorrect option is 2: $b^2-4ac\>0$ would mean the roots are real and distinct (unequal). The incorrect option is 3: $b^2-4ac\<0$ would mean the roots are not real (complex or imaginary). The incorrect option is 4: None of these is incorrect because option 1 correctly describes the condition for real and equal roots.